The Optimal Solution of a Non-Convex State-Dependent LQR Problem and Its Applications
Xu, Xudan; Zhu, J. Jim; Zhang, Ping
2014-01-01
This paper studies a Non-convex State-dependent Linear Quadratic Regulator (NSLQR) problem, in which the control penalty weighting matrix in the performance index is state-dependent. A necessary and sufficient condition for the optimal solution is established with a rigorous proof by Euler-Lagrange Equation. It is found that the optimal solution of the NSLQR problem can be obtained by solving a Pseudo-Differential-Riccati-Equation (PDRE) simultaneously with the closed-loop system equation. A Comparison Theorem for the PDRE is given to facilitate solution methods for the PDRE. A linear time-variant system is employed as an example in simulation to verify the proposed optimal solution. As a non-trivial application, a goal pursuit process in psychology is modeled as a NSLQR problem and two typical goal pursuit behaviors found in human and animals are reproduced using different control weighting . It is found that these two behaviors save control energy and cause less stress over Conventional Control Behavior typified by the LQR control with a constant control weighting , in situations where only the goal discrepancy at the terminal time is of concern, such as in Marathon races and target hitting missions. PMID:24747417
CVXPY: A Python-Embedded Modeling Language for Convex Optimization
Diamond, Steven; Boyd, Stephen
2016-01-01
CVXPY is a domain-specific language for convex optimization embedded in Python. It allows the user to express convex optimization problems in a natural syntax that follows the math, rather than in the restrictive standard form required by solvers. CVXPY makes it easy to combine convex optimization with high-level features of Python such as parallelism and object-oriented design. CVXPY is available at http://www.cvxpy.org/ under the GPL license, along with documentation and examples. PMID:27375369
Robust boosting via convex optimization
NASA Astrophysics Data System (ADS)
Rätsch, Gunnar
2001-12-01
In this work we consider statistical learning problems. A learning machine aims to extract information from a set of training examples such that it is able to predict the associated label on unseen examples. We consider the case where the resulting classification or regression rule is a combination of simple rules - also called base hypotheses. The so-called boosting algorithms iteratively find a weighted linear combination of base hypotheses that predict well on unseen data. We address the following issues: o The statistical learning theory framework for analyzing boosting methods. We study learning theoretic guarantees on the prediction performance on unseen examples. Recently, large margin classification techniques emerged as a practical result of the theory of generalization, in particular Boosting and Support Vector Machines. A large margin implies a good generalization performance. Hence, we analyze how large the margins in boosting are and find an improved algorithm that is able to generate the maximum margin solution. o How can boosting methods be related to mathematical optimization techniques? To analyze the properties of the resulting classification or regression rule, it is of high importance to understand whether and under which conditions boosting converges. We show that boosting can be used to solve large scale constrained optimization problems, whose solutions are well characterizable. To show this, we relate boosting methods to methods known from mathematical optimization, and derive convergence guarantees for a quite general family of boosting algorithms. o How to make Boosting noise robust? One of the problems of current boosting techniques is that they are sensitive to noise in the training sample. In order to make boosting robust, we transfer the soft margin idea from support vector learning to boosting. We develop theoretically motivated regularized algorithms that exhibit a high noise robustness. o How to adapt boosting to regression problems
Design of quasi-phasematching gratings via convex optimization.
Phillips, C R; Gallmann, L; Fejer, M M
2013-04-22
We propose a new approach to quasi-phasematching (QPM) design based on convex optimization. We show that with this approach, globally optimum solutions to several important QPM design problems can be determined. The optimization framework is highly versatile, enabling the user to trade-off different objectives and constraints according to the particular application. The convex problems presented consist of simple objective and constraint functions involving a few thousand variables, and can therefore be solved quite straightforwardly. We consider three examples: (1) synthesis of a target pulse profile via difference frequency generation (DFG) from two ultrashort input pulses, (2) the design of a custom DFG transfer function, and (3) a new approach enabling the suppression of spectral gain narrowing in chirped-QPM-based optical parametric chirped pulse amplification (OPCPA). These examples illustrate the power and versatility of convex optimization in the context of QPM devices. PMID:23609719
Rapid Generation of Optimal Asteroid Powered Descent Trajectories Via Convex Optimization
NASA Technical Reports Server (NTRS)
Pinson, Robin; Lu, Ping
2015-01-01
This paper investigates a convex optimization based method that can rapidly generate the fuel optimal asteroid powered descent trajectory. The ultimate goal is to autonomously design the optimal powered descent trajectory on-board the spacecraft immediately prior to the descent burn. Compared to a planetary powered landing problem, the major difficulty is the complex gravity field near the surface of an asteroid that cannot be approximated by a constant gravity field. This paper uses relaxation techniques and a successive solution process that seeks the solution to the original nonlinear, nonconvex problem through the solutions to a sequence of convex optimal control problems.
Lagrange duality theory for convex control problems
NASA Technical Reports Server (NTRS)
Hager, W. W.; Mitter, S. K.
1976-01-01
The Lagrange dual to a control problem is studied. The principal result based on the Hahn-Banach theorem proves that the dual problem has an optimal solution if there exists an interior point for the constraint set. A complementary slackness condition holds, if the primal problem has an optimal solution. A necessary and sufficient condition for the optimality of solutions to the primal and the dual problem is also presented.
A Deep-Cutting-Plane Technique for Reverse Convex Optimization.
Moshirvaziri, K; Amouzegar, M A
2011-08-01
A large number of problems in engineering design and in many areas of social and physical sciences and technology lend themselves to particular instances of problems studied in this paper. Cutting-plane methods have traditionally been used as an effective tool in devising exact algorithms for solving convex and large-scale combinatorial optimization problems. Its utilization in nonconvex optimization has been also promising. A cutting plane, essentially a hyperplane defined by a linear inequality, can be used to effectively reduce the computational efforts in search of a global solution. Each cut is generated in order to eliminate a large portion of the search domain. Thus, a deep cut is intuitively superior in which it will exclude a larger set of extraneous points from consideration. This paper is concerned with the development of deep-cutting-plane techniques applied to reverse-convex programs. An upper bound and a lower bound for the optimal value are found, updated, and improved at each iteration. The algorithm terminates when the two bounds collapse or all the generated subdivisions have been fathomed. Finally, computational considerations and numerical results on a set of test problems are discussed. An illustrative example, walking through the steps of the algorithm and explaining the computational process, is presented. PMID:21296710
Optimization-based mesh correction with volume and convexity constraints
NASA Astrophysics Data System (ADS)
D'Elia, Marta; Ridzal, Denis; Peterson, Kara J.; Bochev, Pavel; Shashkov, Mikhail
2016-05-01
We consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. This volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach.
Systematization of problems on ball estimates of a convex compactum
NASA Astrophysics Data System (ADS)
Dudov, S. I.
2015-09-01
We consider a class of finite-dimensional problems on the estimation of a convex compactum by a ball of an arbitrary norm in the form of extremal problems whose goal function is expressed via the function of the distance to the farthest point of the compactum and the function of the distance to the nearest point of the compactum or its complement. Special attention is devoted to the problem of estimating (approximating) a convex compactum by a ball of fixed radius in the Hausdorff metric. It is proved that this problem plays the role of the canonical problem: solutions of any problem in the class under consideration can be expressed via solutions of this problem for certain values of the radius. Based on studying and using the properties of solutions of this canonical problem, we obtain ranges of values of the radius in which the canonical problem expresses solutions of the problems on inscribed and circumscribed balls, the problem of uniform estimate by a ball in the Hausdorff metric, the problem of asphericity of a convex body, the problems of spherical shells of the least thickness and of the least volume for the boundary of a convex body. This makes it possible to arrange the problems in increasing order of the corresponding values of the radius. Bibliography: 34 titles.
NASA Astrophysics Data System (ADS)
Geletu, Abebe; Klöppel, Michael; Hoffmann, Armin; Li, Pu
2015-04-01
Chance constrained optimization problems in engineering applications possess highly nonlinear process models and non-convex structures. As a result, solving a nonlinear non-convex chance constrained optimization (CCOPT) problem remains as a challenging task. The major difficulty lies in the evaluation of probability values and gradients of inequality constraints which are nonlinear functions of stochastic variables. This article proposes a novel analytic approximation to improve the tractability of smooth non-convex chance constraints. The approximation uses a smooth parametric function to define a sequence of smooth nonlinear programs (NLPs). The sequence of optimal solutions of these NLPs remains always feasible and converges to the solution set of the CCOPT problem. Furthermore, Karush-Kuhn-Tucker (KKT) points of the approximating problems converge to a subset of KKT points of the CCOPT problem. Another feature of this approach is that it can handle uncertainties with both Gaussian and/or non-Gaussian distributions.
Optimization-based mesh correction with volume and convexity constraints
D'Elia, Marta; Ridzal, Denis; Peterson, Kara J.; Bochev, Pavel; Shashkov, Mikhail
2016-02-24
Here, we consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. Also, this volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problemmore » in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach.« less
Derivative-free generation and interpolation of convex Pareto optimal IMRT plans
NASA Astrophysics Data System (ADS)
Hoffmann, Aswin L.; Siem, Alex Y. D.; den Hertog, Dick; Kaanders, Johannes H. A. M.; Huizenga, Henk
2006-12-01
In inverse treatment planning for intensity-modulated radiation therapy (IMRT), beamlet intensity levels in fluence maps of high-energy photon beams are optimized. Treatment plan evaluation criteria are used as objective functions to steer the optimization process. Fluence map optimization can be considered a multi-objective optimization problem, for which a set of Pareto optimal solutions exists: the Pareto efficient frontier (PEF). In this paper, a constrained optimization method is pursued to iteratively estimate the PEF up to some predefined error. We use the property that the PEF is convex for a convex optimization problem to construct piecewise-linear upper and lower bounds to approximate the PEF from a small initial set of Pareto optimal plans. A derivative-free Sandwich algorithm is presented in which these bounds are used with three strategies to determine the location of the next Pareto optimal solution such that the uncertainty in the estimated PEF is maximally reduced. We show that an intelligent initial solution for a new Pareto optimal plan can be obtained by interpolation of fluence maps from neighbouring Pareto optimal plans. The method has been applied to a simplified clinical test case using two convex objective functions to map the trade-off between tumour dose heterogeneity and critical organ sparing. All three strategies produce representative estimates of the PEF. The new algorithm is particularly suitable for dynamic generation of Pareto optimal plans in interactive treatment planning.
Guaranteed Blind Sparse Spikes Deconvolution via Lifting and Convex Optimization
NASA Astrophysics Data System (ADS)
Chi, Yuejie
2016-06-01
Neural recordings, returns from radars and sonars, images in astronomy and single-molecule microscopy can be modeled as a linear superposition of a small number of scaled and delayed copies of a band-limited or diffraction-limited point spread function, which is either determined by the nature or designed by the users; in other words, we observe the convolution between a point spread function and a sparse spike signal with unknown amplitudes and delays. While it is of great interest to accurately resolve the spike signal from as few samples as possible, however, when the point spread function is not known a priori, this problem is terribly ill-posed. This paper proposes a convex optimization framework to simultaneously estimate the point spread function as well as the spike signal, by mildly constraining the point spread function to lie in a known low-dimensional subspace. By applying the lifting trick, we obtain an underdetermined linear system of an ensemble of signals with joint spectral sparsity, to which atomic norm minimization is applied. Under mild randomness assumptions of the low-dimensional subspace as well as a separation condition of the spike signal, we prove the proposed algorithm, dubbed as AtomicLift, is guaranteed to recover the spike signal up to a scaling factor as soon as the number of samples is large enough. The extension of AtomicLift to handle noisy measurements is also discussed. Numerical examples are provided to validate the effectiveness of the proposed approaches.
NASA Astrophysics Data System (ADS)
Zhang, Xiao-Ming; Ding, Han
2008-11-01
The concept of uncertainty plays an important role in the design of practical mechanical system. The most common methods for solving uncertainty problems are to model the parameters as a random vector. A natural way to handle the randomness is to admit that a given probability density function represents the uncertainty distribution. However, the drawback of this approach is that the probability distribution is difficult to obtain. In this paper, we use the non-probabilistic convex model to deal with the uncertain parameters in which there is no need for probability density functions. Using the convex model theory, a new method to optimize the dynamic response of mechanical system with uncertain parameters is derived. Because the uncertain parameters can be selected as the optimization parameters, the present method can provide more information about the optimization results than those obtained by the deterministic optimization. The present method is implemented for a torsional vibration system. The numerical results show that the method is effective.
Multiband RF pulses with improved performance via convex optimization
NASA Astrophysics Data System (ADS)
Shang, Hong; Larson, Peder E. Z.; Kerr, Adam; Reed, Galen; Sukumar, Subramaniam; Elkhaled, Adam; Gordon, Jeremy W.; Ohliger, Michael A.; Pauly, John M.; Lustig, Michael; Vigneron, Daniel B.
2016-01-01
Selective RF pulses are commonly designed with the desired profile as a low pass filter frequency response. However, for many MRI and NMR applications, the spectrum is sparse with signals existing at a few discrete resonant frequencies. By specifying a multiband profile and releasing the constraint on "don't-care" regions, the RF pulse performance can be improved to enable a shorter duration, sharper transition, or lower peak B1 amplitude. In this project, a framework for designing multiband RF pulses with improved performance was developed based on the Shinnar-Le Roux (SLR) algorithm and convex optimization. It can create several types of RF pulses with multiband magnitude profiles, arbitrary phase profiles and generalized flip angles. The advantage of this framework with a convex optimization approach is the flexible trade-off of different pulse characteristics. Designs for specialized selective RF pulses for balanced SSFP hyperpolarized (HP) 13C MRI, a dualband saturation RF pulse for 1H MR spectroscopy, and a pre-saturation pulse for HP 13C study were developed and tested.
Multiband RF pulses with improved performance via convex optimization.
Shang, Hong; Larson, Peder E Z; Kerr, Adam; Reed, Galen; Sukumar, Subramaniam; Elkhaled, Adam; Gordon, Jeremy W; Ohliger, Michael A; Pauly, John M; Lustig, Michael; Vigneron, Daniel B
2016-01-01
Selective RF pulses are commonly designed with the desired profile as a low pass filter frequency response. However, for many MRI and NMR applications, the spectrum is sparse with signals existing at a few discrete resonant frequencies. By specifying a multiband profile and releasing the constraint on "don't-care" regions, the RF pulse performance can be improved to enable a shorter duration, sharper transition, or lower peak B1 amplitude. In this project, a framework for designing multiband RF pulses with improved performance was developed based on the Shinnar-Le Roux (SLR) algorithm and convex optimization. It can create several types of RF pulses with multiband magnitude profiles, arbitrary phase profiles and generalized flip angles. The advantage of this framework with a convex optimization approach is the flexible trade-off of different pulse characteristics. Designs for specialized selective RF pulses for balanced SSFP hyperpolarized (HP) (13)C MRI, a dualband saturation RF pulse for (1)H MR spectroscopy, and a pre-saturation pulse for HP (13)C study were developed and tested. PMID:26754063
NASA Astrophysics Data System (ADS)
Sumin, M. I.
2014-01-01
A parametric convex programming problem with an operator equality constraint and a finite set of functional inequality constraints is considered in a Hilbert space. The instability of this problem and, as a consequence, the instability of the classical Lagrange principle for it is closely related to its regularity and the subdifferentiability properties of the value function in the optimization problem. A sequential Lagrange principle in nondifferential form is proved for the indicated convex programming problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. It is shown that the classical Lagrange principle in this problem can be naturally treated as a limiting variant of its stable sequential counterpart. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimal control problems and inverse problems is discussed. For two illustrative problems of these kinds, the corresponding stable Lagrange principles are formulated in sequential form.
Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process
Yin, Chuancun; Yuen, Kam Chuen; Shen, Ying
2015-01-01
We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy. PMID:26351655
Sparse representations and convex optimization as tools for LOFAR radio interferometric imaging
NASA Astrophysics Data System (ADS)
Girard, J. N.; Garsden, H.; Starck, J. L.; Corbel, S.; Woiselle, A.; Tasse, C.; McKean, J. P.; Bobin, J.
2015-08-01
Compressed sensing theory is slowly making its way to solve more and more astronomical inverse problems. We address here the application of sparse representations, convex optimization and proximal theory to radio interferometric imaging. First, we expose the theory behind interferometric imaging, sparse representations and convex optimization, and second, we illustrate their application with numerical tests with SASIR, an implementation of the FISTA, a Forward-Backward splitting algorithm hosted in a LOFAR imager. Various tests have been conducted in Garsden et al., 2015. The main results are: i) an improved angular resolution (super resolution of a factor ≈ 2) with point sources as compared to CLEAN on the same data, ii) correct photometry measurements on a field of point sources at high dynamic range and iii) the imaging of extended sources with improved fidelity. SASIR provides better reconstructions (five time less residuals) of the extended emission as compared to CLEAN. With the advent of large radiotelescopes, there is scope for improving classical imaging methods with convex optimization methods combined with sparse representations.
A Localization Method for Multistatic SAR Based on Convex Optimization
2015-01-01
In traditional localization methods for Synthetic Aperture Radar (SAR), the bistatic range sum (BRS) estimation and Doppler centroid estimation (DCE) are needed for the calculation of target localization. However, the DCE error greatly influences the localization accuracy. In this paper, a localization method for multistatic SAR based on convex optimization without DCE is investigated and the influence of BRS estimation error on localization accuracy is analysed. Firstly, by using the information of each transmitter and receiver (T/R) pair and the target in SAR image, the model functions of T/R pairs are constructed. Each model function’s maximum is on the circumference of the ellipse which is the iso-range for its model function’s T/R pair. Secondly, the target function whose maximum is located at the position of the target is obtained by adding all model functions. Thirdly, the target function is optimized based on gradient descent method to obtain the position of the target. During the iteration process, principal component analysis is implemented to guarantee the accuracy of the method and improve the computational efficiency. The proposed method only utilizes BRSs of a target in several focused images from multistatic SAR. Therefore, compared with traditional localization methods for SAR, the proposed method greatly improves the localization accuracy. The effectivity of the localization approach is validated by simulation experiment. PMID:26566031
A Localization Method for Multistatic SAR Based on Convex Optimization.
Zhong, Xuqi; Wu, Junjie; Yang, Jianyu; Sun, Zhichao; Huang, Yuling; Li, Zhongyu
2015-01-01
In traditional localization methods for Synthetic Aperture Radar (SAR), the bistatic range sum (BRS) estimation and Doppler centroid estimation (DCE) are needed for the calculation of target localization. However, the DCE error greatly influences the localization accuracy. In this paper, a localization method for multistatic SAR based on convex optimization without DCE is investigated and the influence of BRS estimation error on localization accuracy is analysed. Firstly, by using the information of each transmitter and receiver (T/R) pair and the target in SAR image, the model functions of T/R pairs are constructed. Each model function's maximum is on the circumference of the ellipse which is the iso-range for its model function's T/R pair. Secondly, the target function whose maximum is located at the position of the target is obtained by adding all model functions. Thirdly, the target function is optimized based on gradient descent method to obtain the position of the target. During the iteration process, principal component analysis is implemented to guarantee the accuracy of the method and improve the computational efficiency. The proposed method only utilizes BRSs of a target in several focused images from multistatic SAR. Therefore, compared with traditional localization methods for SAR, the proposed method greatly improves the localization accuracy. The effectivity of the localization approach is validated by simulation experiment. PMID:26566031
Gálvez, Akemi; Iglesias, Andrés
2013-01-01
Fitting spline curves to data points is a very important issue in many applied fields. It is also challenging, because these curves typically depend on many continuous variables in a highly interrelated nonlinear way. In general, it is not possible to compute these parameters analytically, so the problem is formulated as a continuous nonlinear optimization problem, for which traditional optimization techniques usually fail. This paper presents a new bioinspired method to tackle this issue. In this method, optimization is performed through a combination of two techniques. Firstly, we apply the indirect approach to the knots, in which they are not initially the subject of optimization but precomputed with a coarse approximation scheme. Secondly, a powerful bioinspired metaheuristic technique, the firefly algorithm, is applied to optimization of data parameterization; then, the knot vector is refined by using De Boor's method, thus yielding a better approximation to the optimal knot vector. This scheme converts the original nonlinear continuous optimization problem into a convex optimization problem, solved by singular value decomposition. Our method is applied to some illustrative real-world examples from the CAD/CAM field. Our experimental results show that the proposed scheme can solve the original continuous nonlinear optimization problem very efficiently. PMID:24376380
Ambikapathi, ArulMurugan; Chan, Tsung-Han; Lin, Chia-Hsiang; Yang, Fei-Shih; Chi, Chong-Yung; Wang, Yue
2016-04-01
Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is a powerful imaging modality to study the pharmacokinetics in a suspected cancer/tumor tissue. The pharmacokinetic (PK) analysis of prostate cancer includes the estimation of time activity curves (TACs), and thereby, the corresponding kinetic parameters (KPs), and plays a pivotal role in diagnosis and prognosis of prostate cancer. In this paper, we endeavor to develop a blind source separation algorithm, namely convex-optimization-based KPs estimation (COKE) algorithm for PK analysis based on compartmental modeling of DCE-MRI data, for effective prostate tumor detection and its quantification. The COKE algorithm first identifies the best three representative pixels in the DCE-MRI data, corresponding to the plasma, fast-flow, and slow-flow TACs, respectively. The estimation accuracy of the flux rate constants (FRCs) of the fast-flow and slow-flow TACs directly affects the estimation accuracy of the KPs that provide the cancer and normal tissue distribution maps in the prostate region. The COKE algorithm wisely exploits the matrix structure (Toeplitz, lower triangular, and exponential decay) of the original nonconvex FRCs estimation problem, and reformulates it into two convex optimization problems that can reliably estimate the FRCs. After estimation of the FRCs, the KPs can be effectively estimated by solving a pixel-wise constrained curve-fitting (convex) problem. Simulation results demonstrate the efficacy of the proposed COKE algorithm. The COKE algorithm is also evaluated with DCE-MRI data of four different patients with prostate cancer and the obtained results are consistent with clinical observations. PMID:26292336
Mehrotra, Sanjay; Papp, Dávid
2014-01-01
We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as nonpolynomial moments, can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with nondifferentiable semi-infinite constraints indexed by an infinite dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of our algorithm even in the solution of traditional semi-infinite convex programming problems, whose constraints are differentiable, and are indexed by an index set of low dimension. After the rate of convergence analysis of the cutting surface algorithm, we extend the authors' moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the central cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.
Mehrotra, Sanjay; Papp, Dávid
2014-01-01
We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as nonpolynomial moments, can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum and stochastic programming at the other. Although our primary motivation is to solve distributionally robustmore » optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with nondifferentiable semi-infinite constraints indexed by an infinite dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of our algorithm even in the solution of traditional semi-infinite convex programming problems, whose constraints are differentiable, and are indexed by an index set of low dimension. After the rate of convergence analysis of the cutting surface algorithm, we extend the authors' moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the central cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.« less
Implementation of a Point Algorithm for Real-Time Convex Optimization
NASA Technical Reports Server (NTRS)
Acikmese, Behcet; Motaghedi, Shui; Carson, John
2007-01-01
The primal-dual interior-point algorithm implemented in G-OPT is a relatively new and efficient way of solving convex optimization problems. Given a prescribed level of accuracy, the convergence to the optimal solution is guaranteed in a predetermined, finite number of iterations. G-OPT Version 1.0 is a flight software implementation written in C. Onboard application of the software enables autonomous, real-time guidance and control that explicitly incorporates mission constraints such as control authority (e.g. maximum thrust limits), hazard avoidance, and fuel limitations. This software can be used in planetary landing missions (Mars pinpoint landing and lunar landing), as well as in proximity operations around small celestial bodies (moons, asteroids, and comets). It also can be used in any spacecraft mission for thrust allocation in six-degrees-of-freedom control.
Higher order sensitivity of solutions to convex programming problems without strict complementarity
NASA Technical Reports Server (NTRS)
Malanowski, Kazimierz
1988-01-01
Consideration is given to a family of convex programming problems which depend on a vector parameter. It is shown that the solutions of the problems and the associated Lagrange multipliers are arbitrarily many times directionally differentiable functions of the parameter, provided that the data of the problems are sufficiently regular. The characterizations of the respective derivatives are given.
Exact Convex Relaxation of Optimal Power Flow in Radial Networks
Gan, LW; Li, N; Topcu, U; Low, SH
2015-01-01
The optimal power flow (OPF) problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. It is nonconvex. We prove that a global optimum of OPF can be obtained by solving a second-order cone program, under a mild condition after shrinking the OPF feasible set slightly, for radial power networks. The condition can be checked a priori, and holds for the IEEE 13, 34, 37, 123-bus networks and two real-world networks.
Simulation of stochastic systems via polynomial chaos expansions and convex optimization
NASA Astrophysics Data System (ADS)
Fagiano, Lorenzo; Khammash, Mustafa
2012-09-01
Polynomial chaos expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and nontrivial manipulation of the model, or the computation of large numbers of simulation runs, rendering the approach too time consuming and impracticable for applications with more than a handful of random variables. We introduce a computationally tractable technique for computing the coefficients of polynomial chaos expansions. The approach exploits a regularization technique with a particular choice of weighting matrices, which allows to take into account the specific features of polynomial chaos expansions. The method, completely based on convex optimization, can be applied to problems with a large number of random variables and uses a modest number of Monte Carlo simulations, while avoiding model manipulations. Additional information on the stochastic process, when available, can be also incorporated in the approach by means of convex constraints. We show the effectiveness of the proposed technique in three applications in diverse fields, including the analysis of a nonlinear electric circuit, a chaotic model of organizational behavior, and finally a chemical oscillator.
Yatabe, Kohei; Oikawa, Yasuhiro
2016-06-10
The windowed Fourier filtering (WFF), defined as a thresholding operation in the windowed Fourier transform (WFT) domain, is a successful method for denoising a phase map and analyzing a fringe pattern. However, it has some shortcomings, such as extremely high redundancy, which results in high computational cost, and difficulty in selecting an appropriate window size. In this paper, an extension of WFF for denoising a wrapped-phase map is proposed. It is formulated as a convex optimization problem using Gabor frames instead of WFT. Two Gabor frames with differently sized windows are used simultaneously so that the above-mentioned issues are resolved. In addition, a differential operator is combined with a Gabor frame in order to preserve discontinuity of the underlying phase map better. Some numerical experiments demonstrate that the proposed method is able to reconstruct a wrapped-phase map, even for a severely contaminated situation. PMID:27409020
The role of convexity for solving some shortest path problems in plane without triangulation
NASA Astrophysics Data System (ADS)
An, Phan Thanh; Hai, Nguyen Ngoc; Hoai, Tran Van
2013-09-01
Solving shortest path problems inside simple polygons is a very classical problem in motion planning. To date, it has usually relied on triangulation of the polygons. The question: "Can one devise a simple O(n) time algorithm for computing the shortest path between two points in a simple polygon (with n vertices), without resorting to a (complicated) linear-time triangulation algorithm?" raised by J. S. B. Mitchell in Handbook of Computational Geometry (J. Sack and J. Urrutia, eds., Elsevier Science B.V., 2000), is still open. The aim of this paper is to show that convexity contributes to the design of efficient algorithms for solving some versions of shortest path problems (namely, computing the convex hull of a finite set of points and convex rope on rays in 2D, computing approximate shortest path between two points inside a simple polygon) without triangulation on the entire polygons. New algorithms are implemented in C and numerical examples are presented.
A Class of Prediction-Correction Methods for Time-Varying Convex Optimization
NASA Astrophysics Data System (ADS)
Simonetto, Andrea; Mokhtari, Aryan; Koppel, Alec; Leus, Geert; Ribeiro, Alejandro
2016-09-01
This paper considers unconstrained convex optimization problems with time-varying objective functions. We propose algorithms with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and correction steps, while sampling the problem data at a constant rate of $1/h$, where $h$ is the length of the sampling interval. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions. The correction step adjusts for the distance between the current prediction and the optimizer at each time step, and consists either of one or multiple gradient steps or Newton steps, which respectively correspond to the gradient trajectory tracking (GTT) or Newton trajectory tracking (NTT) algorithms. Under suitable conditions, we establish that the asymptotic error incurred by both proposed methods behaves as $O(h^2)$, and in some cases as $O(h^4)$, which outperforms the state-of-the-art error bound of $O(h)$ for correction-only methods in the gradient-correction step. Moreover, when the characteristics of the objective function variation are not available, we propose approximate gradient and Newton tracking algorithms (AGT and ANT, respectively) that still attain these asymptotical error bounds. Numerical simulations demonstrate the practical utility of the proposed methods and that they improve upon existing techniques by several orders of magnitude.
libCreme: An optimization library for evaluating convex-roof entanglement measures
NASA Astrophysics Data System (ADS)
Röthlisberger, Beat; Lehmann, Jörg; Loss, Daniel
2012-01-01
We present the software library libCreme which we have previously used to successfully calculate convex-roof entanglement measures of mixed quantum states appearing in realistic physical systems. Evaluating the amount of entanglement in such states is in general a non-trivial task requiring to solve a highly non-linear complex optimization problem. The algorithms provided here are able to achieve to do this for a large and important class of entanglement measures. The library is mostly written in the MATLAB programming language, but is fully compatible to the free and open-source OCTAVE platform. Some inefficient subroutines are written in C/C++ for better performance. This manuscript discusses the most important theoretical concepts and workings of the algorithms, focusing on the actual implementation and usage within the library. Detailed examples in the end should make it easy for the user to apply libCreme to specific problems. Program summaryProgram title:libCreme Catalogue identifier: AEKD_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEKD_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU GPL version 3 No. of lines in distributed program, including test data, etc.: 4323 No. of bytes in distributed program, including test data, etc.: 70 542 Distribution format: tar.gz Programming language: Matlab/Octave and C/C++ Computer: All systems running Matlab or Octave Operating system: All systems running Matlab or Octave Classification: 4.9, 4.15 Nature of problem: Evaluate convex-roof entanglement measures. This involves solving a non-linear (unitary) optimization problem. Solution method: Two algorithms are provided: A conjugate-gradient method using a differential-geometric approach and a quasi-Newton method together with a mapping to Euclidean space. Running time: Typically seconds to minutes for a density matrix of a few low-dimensional systems and a decent implementation of the
NASA Astrophysics Data System (ADS)
Gutiérrez, César; Jiménez, Bienvenido; Novo, Vicente
2005-10-01
In this work we obtain a chain rule for the approximate subdifferential considering a vector-valued proper convex function and its post-composition with a proper convex function of several variables nondecreasing in the sense of the Pareto order. We derive an interesting formula for the conjugate of a composition in the same framework and we prove the chain rule using this formula. To get the results, we require qualification conditions since, in the composition, the initial function is extended vector-valued. This chain rule extends analogous well-known calculus rules obtained when the functions involved are finite and it gives a complementary simple expression for other chain rules proved without assuming any qualification condition. As application we deduce the well-known calculus rule for the addition and we extend the formula for the maximum of functions. Finally, we use them and a scalarization process to obtain Kuhn-Tucker type necessary and sufficient conditions for approximate solutions in convex Pareto problems. These conditions extend other obtained in scalar optimization problems.
SLOPE—ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION
Bogdan, Małgorzata; van den Berg, Ewout; Sabatti, Chiara; Su, Weijie; Candès, Emmanuel J.
2015-01-01
We introduce a new estimator for the vector of coefficients β in the linear model y = Xβ + z, where X has dimensions n × p with p possibly larger than n. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to minb∈ℝp12‖y−Xb‖ℓ22+λ1|b|(1)+λ2|b|(2)+⋯+λp|b|(p),where λ1 ≥ λ2 ≥ … ≥ λp ≥ 0 and |b|(1)≥|b|(2)≥⋯≥|b|(p) are the decreasing absolute values of the entries of b. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ℓ1 procedures such as the Lasso. Here, the regularizer is a sorted ℓ1 norm, which penalizes the regression coefficients according to their rank: the higher the rank—that is, stronger the signal—the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] procedure (BH) which compares more significant p-values with more stringent thresholds. One notable choice of the sequence {λi} is given by the BH critical values λBH(i)=z(1−i⋅q/2p), where q ∈ (0, 1) and z(α) is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λBH provably controls FDR at level q. Moreover, it also appears to have appreciable inferential properties under more general designs X while having substantial power, as demonstrated in a series of experiments running on both simulated and real data. PMID:26709357
Wang, Li; Gao, Yaozong; Shi, Feng; Liao, Shu; Li, Gang; Chen, Ken Chung; Shen, Steve G. F.; Yan, Jin; Lee, Philip K. M.; Chow, Ben; Liu, Nancy X.; Xia, James J.; Shen, Dinggang
2014-04-15
Purpose: Cone-beam computed tomography (CBCT) is an increasingly utilized imaging modality for the diagnosis and treatment planning of the patients with craniomaxillofacial (CMF) deformities. Accurate segmentation of CBCT image is an essential step to generate three-dimensional (3D) models for the diagnosis and treatment planning of the patients with CMF deformities. However, due to the poor image quality, including very low signal-to-noise ratio and the widespread image artifacts such as noise, beam hardening, and inhomogeneity, it is challenging to segment the CBCT images. In this paper, the authors present a new automatic segmentation method to address these problems. Methods: To segment CBCT images, the authors propose a new method for fully automated CBCT segmentation by using patch-based sparse representation to (1) segment bony structures from the soft tissues and (2) further separate the mandible from the maxilla. Specifically, a region-specific registration strategy is first proposed to warp all the atlases to the current testing subject and then a sparse-based label propagation strategy is employed to estimate a patient-specific atlas from all aligned atlases. Finally, the patient-specific atlas is integrated into amaximum a posteriori probability-based convex segmentation framework for accurate segmentation. Results: The proposed method has been evaluated on a dataset with 15 CBCT images. The effectiveness of the proposed region-specific registration strategy and patient-specific atlas has been validated by comparing with the traditional registration strategy and population-based atlas. The experimental results show that the proposed method achieves the best segmentation accuracy by comparison with other state-of-the-art segmentation methods. Conclusions: The authors have proposed a new CBCT segmentation method by using patch-based sparse representation and convex optimization, which can achieve considerably accurate segmentation results in CBCT
Wang, Li; Chen, Ken Chung; Gao, Yaozong; Shi, Feng; Liao, Shu; Li, Gang; Shen, Steve G. F.; Yan, Jin; Lee, Philip K. M.; Chow, Ben; Liu, Nancy X.; Xia, James J.; Shen, Dinggang
2014-01-01
Purpose: Cone-beam computed tomography (CBCT) is an increasingly utilized imaging modality for the diagnosis and treatment planning of the patients with craniomaxillofacial (CMF) deformities. Accurate segmentation of CBCT image is an essential step to generate three-dimensional (3D) models for the diagnosis and treatment planning of the patients with CMF deformities. However, due to the poor image quality, including very low signal-to-noise ratio and the widespread image artifacts such as noise, beam hardening, and inhomogeneity, it is challenging to segment the CBCT images. In this paper, the authors present a new automatic segmentation method to address these problems. Methods: To segment CBCT images, the authors propose a new method for fully automated CBCT segmentation by using patch-based sparse representation to (1) segment bony structures from the soft tissues and (2) further separate the mandible from the maxilla. Specifically, a region-specific registration strategy is first proposed to warp all the atlases to the current testing subject and then a sparse-based label propagation strategy is employed to estimate a patient-specific atlas from all aligned atlases. Finally, the patient-specific atlas is integrated into a maximum a posteriori probability-based convex segmentation framework for accurate segmentation. Results: The proposed method has been evaluated on a dataset with 15 CBCT images. The effectiveness of the proposed region-specific registration strategy and patient-specific atlas has been validated by comparing with the traditional registration strategy and population-based atlas. The experimental results show that the proposed method achieves the best segmentation accuracy by comparison with other state-of-the-art segmentation methods. Conclusions: The authors have proposed a new CBCT segmentation method by using patch-based sparse representation and convex optimization, which can achieve considerably accurate segmentation results in CBCT
Non-parametric Single View Reconstruction of Curved Objects Using Convex Optimization
NASA Astrophysics Data System (ADS)
Oswald, Martin R.; Töppe, Eno; Kolev, Kalin; Cremers, Daniel
We propose a convex optimization framework delivering intuitive and reasonable 3D meshes from a single photograph. For a given input image, the user can quickly obtain a segmentation of the object in question. Our algorithm then automatically generates an admissible closed surface of arbitrary topology without the requirement of tedious user input. Moreover we provide a tool by which the user is able to interactively modify the result afterwards through parameters and simple operations in a 2D image space. The algorithm targets a limited but relevant class of real world objects. The object silhouette and the additional user input enter a functional which can be optimized globally in a few seconds using recently developed convex relaxation techniques parallelized on state-of-the-art graphics hardware.
Smoothers for Optimization Problems
NASA Technical Reports Server (NTRS)
Arian, Eyal; Ta'asan, Shlomo
1996-01-01
We present a multigrid one-shot algorithm, and a smoothing analysis, for the numerical solution of optimal control problems which are governed by an elliptic PDE. The analysis provides a simple tool to determine a smoothing minimization process which is essential for multigrid application. Numerical results include optimal control of boundary data using different discretization schemes and an optimal shape design problem in 2D with Dirichlet boundary conditions.
ERIC Educational Resources Information Center
Scott, Paul
2006-01-01
A "convex" polygon is one with no re-entrant angles. Alternatively one can use the standard convexity definition, asserting that for any two points of the convex polygon, the line segment joining them is contained completely within the polygon. In this article, the author provides a solution to a problem involving convex lattice polygons.
Razavi, Sonia M; Gonzalez, Marcial; Cuitiño, Alberto M
2015-04-30
We propose a general framework for determining optimal relationships for tensile strength of doubly convex tablets under diametrical compression. This approach is based on the observation that tensile strength is directly proportional to the breaking force and inversely proportional to a non-linear function of geometric parameters and materials properties. This generalization reduces to the analytical expression commonly used for flat faced tablets, i.e., Hertz solution, and to the empirical relationship currently used in the pharmaceutical industry for convex-faced tablets, i.e., Pitt's equation. Under proper parametrization, optimal tensile strength relationship can be determined from experimental results by minimizing a figure of merit of choice. This optimization is performed under the first-order approximation that a flat faced tablet and a doubly curved tablet have the same tensile strength if they have the same relative density and are made of the same powder, under equivalent manufacturing conditions. Furthermore, we provide a set of recommendations and best practices for assessing the performance of optimal tensile strength relationships in general. Based on these guidelines, we identify two new models, namely the general and mechanistic models, which are effective and predictive alternatives to the tensile strength relationship currently used in the pharmaceutical industry. PMID:25683146
Reduced Complexity HMM Filtering With Stochastic Dominance Bounds: A Convex Optimization Approach
NASA Astrophysics Data System (ADS)
Krishnamurthy, Vikram; Rojas, Cristian R.
2014-12-01
This paper uses stochastic dominance principles to construct upper and lower sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using convex optimization methods for nuclear norm minimization with copositive constraints, we construct low rank stochastic marices so that the optimal filters using these matrices provably lower and upper bound (with respect to a partially ordered set) the true filtered distribution at each time instant. Since these matrices are low rank (say R), the computational cost of evaluating the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance sampling filter is presented that exploits these upper and lower bounds to estimate the optimal posterior. Finally, using the Dobrushin coefficient, explicit bounds are given on the variational norm between the true posterior and the upper and lower bounds.
Wang, Qin; Zhang, Bo; Wang, Muhua; Wu, Jiang; Li, Yuyou; Gao, Yingxin; Li, Weicheng; Jin, Yong
2016-11-01
The batch and column experimental studies on the adsorption of phosphate onto synthetic lepidocrocite from reclaimed water are presented. A second-order polynomial model in the batch study is successfully applied to describe phosphate immobilization performance using the response surface methodology. The model proposed is further linked with the convex optimization method to determine the optimal variables for maximum phosphate uptake since convex method is a global optimization method. Consequently, under optimal parameters determined as pH of 3.88, an initial P concentration of 0.66 mg/L, and a dosage of 0.15 g, the corresponding phosphate removal efficiency can reach up to 97.4%. Adsorption behavior is further revealed by X-ray photoelectron spectroscopy observation and FTIR spectra. A comparative column study indicates that co-existing competing anions in artificial reclaimed water do not significantly interfere with P adsorption under the neutral condition. The experimental results highlight that synthetic lepidocrocite is an excellent absorbent for sustainable P removal from reclaimed water. PMID:27121116
Nonsmooth Neural Network for Convex Time-Dependent Constraint Satisfaction Problems.
Di Marco, Mauro; Forti, Mauro; Nistri, Paolo; Pancioni, Luca
2016-02-01
This paper introduces a nonsmooth (NS) neural network that is able to operate in a time-dependent (TD) context and is potentially useful for solving some classes of NS-TD problems. The proposed network is named nonsmooth time-dependent network (NTN) and is an extension to a TD setting of a previous NS neural network for programming problems. Suppose C(t), t ≥ 0, is a nonempty TD convex feasibility set defined by TD inequality constraints. The constraints are in general NS (nondifferentiable) functions of the state variables and time. NTN is described by the subdifferential with respect to the state variables of an NS-TD barrier function and a vector field corresponding to the unconstrained dynamics. This paper shows that for suitable values of the penalty parameter, the NTN dynamics displays two main phases. In the first phase, any solution of NTN not starting in C(0) at t=0 is able to reach the moving set C(·) in finite time th , whereas in the second phase, the solution tracks the moving set, i.e., it stays within C(t) for all subsequent times t ≥ t(h). NTN is thus able to find an exact feasible solution in finite time and also to provide an exact feasible solution for subsequent times. This new and peculiar dynamics displayed by NTN is potentially useful for addressing some significant TD signal processing tasks. As an illustration, this paper discusses a number of examples where NTN is applied to the solution of NS-TD convex feasibility problems. PMID:25769174
Rapid Generation of Optimal Asteroid Powered Descent Trajectories Via Convex Optimization
NASA Technical Reports Server (NTRS)
Pinson, Robin; Lu, Ping
2015-01-01
Mission proposals that land on asteroids are becoming popular. However, in order to have a successful mission the spacecraft must reliably and softly land at the intended landing site. The problem under investigation is how to design a fuel-optimal powered descent trajectory that can be quickly computed on-board the spacecraft, without interaction from ground control. An optimal trajectory designed immediately prior to the descent burn has many advantages. These advantages include the ability to use the actual vehicle starting state as the initial condition in the trajectory design and the ease of updating the landing target site if the original landing site is no longer viable. For long trajectories, the trajectory can be updated periodically by a redesign of the optimal trajectory based on current vehicle conditions to improve the guidance performance. One of the key drivers for being completely autonomous is the infrequent and delayed communication between ground control and the vehicle. Challenges that arise from designing an asteroid powered descent trajectory include complicated nonlinear gravity fields, small rotating bodies and low thrust vehicles.
Fast inference of ill-posed problems within a convex space
NASA Astrophysics Data System (ADS)
Fernandez-de-Cossio-Diaz, J.; Mulet, R.
2016-07-01
In multiple scientific and technological applications we face the problem of having low dimensional data to be justified by a linear model defined in a high dimensional parameter space. The difference in dimensionality makes the problem ill-defined: the model is consistent with the data for many values of its parameters. The objective is to find the probability distribution of parameter values consistent with the data, a problem that can be cast as the exploration of a high dimensional convex polytope. In this work we introduce a novel algorithm to solve this problem efficiently. It provides results that are statistically indistinguishable from currently used numerical techniques while its running time scales linearly with the system size. We show that the algorithm performs robustly in many abstract and practical applications. As working examples we simulate the effects of restricting reaction fluxes on the space of feasible phenotypes of a genome scale Escherichia coli metabolic network and infer the traffic flow between origin and destination nodes in a real communication network.
Optimal structural design via optimality criteria as a nonsmooth mechanics problem
NASA Astrophysics Data System (ADS)
Tzaferopoulos, M. Ap.; Stravroulakis, G. E.
1995-06-01
In the theory of plastic structural design via optimality criteria (due to W. Prager), the optimal design problem is transformed to a nonlinear elastic structural analysis problem with appropriate stress-strain laws, which generally include complete vertical branches. In this context, the concept of structural universe (in the sense of G. Rozvany) permits the treatment of complicated optimal layout problems. Recent progress in the field of nonsmooth mechanics makes the solution of structural analysis problems with this kind of 'complete' law possible. Elements from the two fields are combined in this paper for the solution of optimal design and layout problems for structures. The optimal layout of plane trusses with various specific cost functions is studied here as a representative problem. The use of convex, continuous and piecewise linear specific cost functions for the structural members leads to problems of linear variational inequalities or equivalently piecewise linear, convex but nonsmooth optimization problems, which are solved by means of an iterative algorithm based on sequential linear programming techniques. Numerical examples illustrate the theory and its applicability to practical engineering structures. Following a parametric investigation of an optimal bridge design, certain aspects of the optimal truss layout problem are discussed, which can be extended to other types of structural systems as well.
NASA Astrophysics Data System (ADS)
Chen, Shibing; Wang, Xu-Jia
2016-01-01
In this paper we prove the strict c-convexity and the C 1, α regularity for potential functions in optimal transportation under condition (A3w). These results were obtained by Caffarelli [1,3,4] for the cost c (x, y) =| x - y | 2, by Liu [11], Loeper [15], Trudinger and Wang [20] for costs satisfying the condition (A3). For costs satisfying the condition (A3w), the results have also been proved by Figalli, Kim, and McCann [6], assuming that the initial and target domains are uniformly c-convex, see also [21]; and by Guillen and Kitagawa [8], assuming the cost function satisfies A3w in larger domains. In this paper we prove the strict c-convexity and the C 1, α regularity assuming either the support of source density is compactly contained in a larger domain where the cost function satisfies A3w, or the dimension 2 ≤ n ≤ 4.
NASA Astrophysics Data System (ADS)
Hoffmann, Aswin L.; den Hertog, Dick; Siem, Alex Y. D.; Kaanders, Johannes H. A. M.; Huizenga, Henk
2008-11-01
Finding fluence maps for intensity-modulated radiation therapy (IMRT) can be formulated as a multi-criteria optimization problem for which Pareto optimal treatment plans exist. To account for the dose-per-fraction effect of fractionated IMRT, it is desirable to exploit radiobiological treatment plan evaluation criteria based on the linear-quadratic (LQ) cell survival model as a means to balance the radiation benefits and risks in terms of biologic response. Unfortunately, the LQ-model-based radiobiological criteria are nonconvex functions, which make the optimization problem hard to solve. We apply the framework proposed by Romeijn et al (2004 Phys. Med. Biol. 49 1991-2013) to find transformations of LQ-model-based radiobiological functions and establish conditions under which transformed functions result in equivalent convex criteria that do not change the set of Pareto optimal treatment plans. The functions analysed are: the LQ-Poisson-based model for tumour control probability (TCP) with and without inter-patient heterogeneity in radiation sensitivity, the LQ-Poisson-based relative seriality s-model for normal tissue complication probability (NTCP), the equivalent uniform dose (EUD) under the LQ-Poisson model and the fractionation-corrected Probit-based model for NTCP according to Lyman, Kutcher and Burman. These functions differ from those analysed before in that they cannot be decomposed into elementary EUD or generalized-EUD functions. In addition, we show that applying increasing and concave transformations to the convexified functions is beneficial for the piecewise approximation of the Pareto efficient frontier.
Cai, Ailong; Wang, Linyuan; Yan, Bin; Li, Lei; Zhang, Hanming; Hu, Guoen
2015-10-01
An efficient iterative algorithm, based on recent work in non-convex optimization and generalized p-shrinkage mappings, is proposed for volume image reconstruction from circular cone-beam scans. Conventional total variation regularization makes use of L1 norm of gradient magnitude images (GMI). However, this paper utilizes a generalized penalty function, induced by p-shrinkage, of GMI which is proven to be a better measurement of its sparsity. The reconstruction model is formed using generalized total p-variation (TpV) minimization, which differs with the state of the art methods, with the constraint that the estimated projection data is within a specified tolerance of the available data and that the values of the volume image are non-negative. Theoretically, the proximal mapping for penalty functions induced by p-shrinkage has an exact and closed-form expression; thus, the constrained optimization can be stably and efficiently solved by the alternating direction minimization (ADM) scheme. Each sub-problem decoupled by variable splitting is minimized by explicit and easy-to-implement formulas developed by ADM. The proposed algorithm is efficiently implemented using a graphics processing unit and is referred to as "TpV-ADM." This method is robust and accurate even for very few view reconstruction datasets. Verifications and comparisons performed using various datasets (including ideal, noisy, and real projections) illustrate that the proposed method is effective and promising. PMID:26233922
Class and Home Problems: Optimization Problems
ERIC Educational Resources Information Center
Anderson, Brian J.; Hissam, Robin S.; Shaeiwitz, Joseph A.; Turton, Richard
2011-01-01
Optimization problems suitable for all levels of chemical engineering students are available. These problems do not require advanced mathematical techniques, since they can be solved using typical software used by students and practitioners. The method used to solve these problems forces students to understand the trends for the different terms…
Convex optimization of MRI exposure for mitigation of RF-heating from active medical implants
NASA Astrophysics Data System (ADS)
Córcoles, Juan; Zastrow, Earl; Kuster, Niels
2015-09-01
Local RF-heating of elongated medical implants during magnetic resonance imaging (MRI) may pose a significant health risk to patients. The actual patient risk depends on various parameters including RF magnetic field strength and frequency, MR coil design, patient’s anatomy, posture, and imaging position, implant location, RF coupling efficiency of the implant, and the bio-physiological responses associated with the induced local heating. We present three constrained convex optimization strategies that incorporate the implant’s RF-heating characteristics, for the reduction of local heating of medical implants during MRI. The study emphasizes the complementary performances of the different formulations. The analysis demonstrates that RF-induced heating of elongated metallic medical implants can be carefully controlled and balanced against MRI quality. A reduction of heating of up to 25 dB can be achieved at the cost of reduced uniformity in the magnitude of the B1+ field of less than 5%. The current formulations incorporate a priori knowledge of clinically-specific parameters, which is assumed to be available. Before these techniques can be applied practically in the broader clinical context, further investigations are needed to determine whether reduced access to a priori knowledge regarding, e.g. the patient’s anatomy, implant routing, RF-transmitter, and RF-implant coupling, can be accepted within reasonable levels of uncertainty.
Convex optimization of MRI exposure for mitigation of RF-heating from active medical implants.
Córcoles, Juan; Zastrow, Earl; Kuster, Niels
2015-09-21
Local RF-heating of elongated medical implants during magnetic resonance imaging (MRI) may pose a significant health risk to patients. The actual patient risk depends on various parameters including RF magnetic field strength and frequency, MR coil design, patient's anatomy, posture, and imaging position, implant location, RF coupling efficiency of the implant, and the bio-physiological responses associated with the induced local heating. We present three constrained convex optimization strategies that incorporate the implant's RF-heating characteristics, for the reduction of local heating of medical implants during MRI. The study emphasizes the complementary performances of the different formulations. The analysis demonstrates that RF-induced heating of elongated metallic medical implants can be carefully controlled and balanced against MRI quality. A reduction of heating of up to 25 dB can be achieved at the cost of reduced uniformity in the magnitude of the B(1)(+) field of less than 5%. The current formulations incorporate a priori knowledge of clinically-specific parameters, which is assumed to be available. Before these techniques can be applied practically in the broader clinical context, further investigations are needed to determine whether reduced access to a priori knowledge regarding, e.g. the patient's anatomy, implant routing, RF-transmitter, and RF-implant coupling, can be accepted within reasonable levels of uncertainty. PMID:26350025
A Problem on Optimal Transportation
ERIC Educational Resources Information Center
Cechlarova, Katarina
2005-01-01
Mathematical optimization problems are not typical in the classical curriculum of mathematics. In this paper we show how several generalizations of an easy problem on optimal transportation were solved by gifted secondary school pupils in a correspondence mathematical seminar, how they can be used in university courses of linear programming and…
Automatic Treatment Planning with Convex Imputing
NASA Astrophysics Data System (ADS)
Sayre, G. A.; Ruan, D.
2014-03-01
Current inverse optimization-based treatment planning for radiotherapy requires a set of complex DVH objectives to be simultaneously minimized. This process, known as multi-objective optimization, is challenging due to non-convexity in individual objectives and insufficient knowledge in the tradeoffs among the objective set. As such, clinical practice involves numerous iterations of human intervention that is costly and often inconsistent. In this work, we propose to address treatment planning with convex imputing, a new-data mining technique that explores the existence of a latent convex objective whose optimizer reflects the DVH and dose-shaping properties of previously optimized cases. Using ten clinical prostate cases as the basis for comparison, we imputed a simple least-squares problem from the optimized solutions of the prostate cases, and show that the imputed plans are more consistent than their clinical counterparts in achieving planning goals.
NASA Astrophysics Data System (ADS)
Chen, Jian; Matuttis, Hans-Georg
2013-02-01
We report our experiences with the optimization and parallelization of a discrete element code for convex polyhedra on multi-core machines and introduce a novel variant of the sort-and-sweep neighborhood algorithm. While in theory the whole code in itself parallelizes ideally, in practice the results on different architectures with different compilers and performance measurement tools depend very much on the particle number and optimization of the code. After difficulties with the interpretation of the data for speedup and efficiency are overcome, respectable parallelization speedups could be obtained.
A near-optimal heuristic for minimum weight triangulation of convex polygons
Levcopoulos, C.; Krznaric, D.
1997-06-01
A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + {epsilon} from the optimum, where {epsilon} is an arbitrarily small positive constant. This is the first sub-cubic algorithm which guarantees such an approximation factor, and it has interesting applications.
Models for optimal harvest with convex function of growth rate of a population
Lyashenko, O.I.
1995-12-10
Two models for growth of a population, which are described by a Cauchy problem for an ordinary differential equation with right-hand side depending on the population size and time, are investigated. The first model is time-discrete, i.e., the moments of harvest are fixed and discrete. The second model is time-continuous, i.e., a crop is harvested continuously in time. For autonomous systems, the second model is a particular case of the variational model for optimal control with constraints investigated in. However, the prerequisites and the method of investigation are somewhat different, for they are based on Lemma 1 presented below. In this paper, the existence and uniqueness theorem for the solution of the discrete and continuous problems of optimal harvest is proved, and the corresponding algorithms are presented. The results obtained are illustrated by a model for growth of the light-requiring green alga Chlorella.
Lossless Convexification of Control Constraints for a Class of Nonlinear Optimal Control Problems
NASA Technical Reports Server (NTRS)
Blackmore, Lars; Acikmese, Behcet; Carson, John M.,III
2012-01-01
In this paper we consider a class of optimal control problems that have continuous-time nonlinear dynamics and nonconvex control constraints. We propose a convex relaxation of the nonconvex control constraints, and prove that the optimal solution to the relaxed problem is the globally optimal solution to the original problem with nonconvex control constraints. This lossless convexification enables a computationally simpler problem to be solved instead of the original problem. We demonstrate the approach in simulation with a planetary soft landing problem involving a nonlinear gravity field.
Convex Regression with Interpretable Sharp Partitions
Petersen, Ashley; Simon, Noah; Witten, Daniela
2016-01-01
We consider the problem of predicting an outcome variable on the basis of a small number of covariates, using an interpretable yet non-additive model. We propose convex regression with interpretable sharp partitions (CRISP) for this task. CRISP partitions the covariate space into blocks in a data-adaptive way, and fits a mean model within each block. Unlike other partitioning methods, CRISP is fit using a non-greedy approach by solving a convex optimization problem, resulting in low-variance fits. We explore the properties of CRISP, and evaluate its performance in a simulation study and on a housing price data set.
Robust Utility Maximization Under Convex Portfolio Constraints
Matoussi, Anis; Mezghani, Hanen Mnif, Mohamed
2015-04-15
We study a robust maximization problem from terminal wealth and consumption under a convex constraints on the portfolio. We state the existence and the uniqueness of the consumption–investment strategy by studying the associated quadratic backward stochastic differential equation. We characterize the optimal control by using the duality method and deriving a dynamic maximum principle.
Interval-Valued Optimization Problems Involving (α, ρ)-Right Upper-Dini-Derivative Functions
2014-01-01
We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals. PMID:24982989
Interval-valued optimization problems involving (α, ρ)-right upper-Dini-derivative functions.
Preda, Vasile
2014-01-01
We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals. PMID:24982989
Optimization and geophysical inverse problems
Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F.; Gill, P.; Heinkenschloss, M.; Johnson, L.; McEvilly, T.; More, J.; Newman, G.; Oldenburg, D.; Parker, P.; Porto, B.; Sen, M.; Torczon, V.; Vasco, D.; Woodward, N.B.
2000-10-01
A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a set of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the objective function often incorporates some additional form of regularization, such as a measure of smoothness
Convex Formulations of Learning from Crowds
NASA Astrophysics Data System (ADS)
Kajino, Hiroshi; Kashima, Hisashi
It has attracted considerable attention to use crowdsourcing services to collect a large amount of labeled data for machine learning, since crowdsourcing services allow one to ask the general public to label data at very low cost through the Internet. The use of crowdsourcing has introduced a new challenge in machine learning, that is, coping with low quality of crowd-generated data. There have been many recent attempts to address the quality problem of multiple labelers, however, there are two serious drawbacks in the existing approaches, that are, (i) non-convexity and (ii) task homogeneity. Most of the existing methods consider true labels as latent variables, which results in non-convex optimization problems. Also, the existing models assume only single homogeneous tasks, while in realistic situations, clients can offer multiple tasks to crowds and crowd workers can work on different tasks in parallel. In this paper, we propose a convex optimization formulation of learning from crowds by introducing personal models of individual crowds without estimating true labels. We further extend the proposed model to multi-task learning based on the resemblance between the proposed formulation and that for an existing multi-task learning model. We also devise efficient iterative methods for solving the convex optimization problems by exploiting conditional independence structures in multiple classifiers.
Approximating random quantum optimization problems
NASA Astrophysics Data System (ADS)
Hsu, B.; Laumann, C. R.; Läuchli, A. M.; Moessner, R.; Sondhi, S. L.
2013-06-01
We report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem k-body quantum satisfiability (k-QSAT) on large random graphs. As an approximation strategy, we optimize the solution space over “classical” product states, which in turn introduces a novel autonomous classical optimization problem, PSAT, over a space of continuous degrees of freedom rather than discrete bits. Our central results are (i) the derivation of a set of bounds and approximations in various limits of the problem, several of which we believe may be amenable to a rigorous treatment; (ii) a demonstration that an approximation based on a greedy algorithm borrowed from the study of frustrated magnetism performs well over a wide range in parameter space, and its performance reflects the structure of the solution space of random k-QSAT. Simulated annealing exhibits metastability in similar “hard” regions of parameter space; and (iii) a generalization of belief propagation algorithms introduced for classical problems to the case of continuous spins. This yields both approximate solutions, as well as insights into the free energy “landscape” of the approximation problem, including a so-called dynamical transition near the satisfiability threshold. Taken together, these results allow us to elucidate the phase diagram of random k-QSAT in a two-dimensional energy-density-clause-density space.
NASA Astrophysics Data System (ADS)
da Gama, Rogério Martins Saldanha
2015-10-01
In this paper, we study the steady-state (coupled) conduction-radiation heat transfer phenomenon in a non-convex opaque blackbody with temperature-dependent thermal conductivity. The mathematical description consists of a nonlinear partial differential equation subjected to a nonlinear boundary condition involving an integral operator that is inherently associated with the non-convexity of the body. The unknown is the absolute temperature distribution. The problem is rewritten with the aid of a Kirchhoff transformation, giving rise to linear partial differential equation and a new unknown. An iterative procedure is proposed for constructing the solution of the problem by means of a sequence of problems, each of them with an equivalent minimum principle. Proofs of convergence as well as existence and uniqueness of the solution are presented. An error estimate, for each element of the sequence, is presented too.
Applying optimization software libraries to engineering problems
NASA Technical Reports Server (NTRS)
Healy, M. J.
1984-01-01
Nonlinear programming, preliminary design problems, performance simulation problems trajectory optimization, flight computer optimization, and linear least squares problems are among the topics covered. The nonlinear programming applications encountered in a large aerospace company are a real challenge to those who provide mathematical software libraries and consultation services. Typical applications include preliminary design studies, data fitting and filtering, jet engine simulations, control system analysis, and trajectory optimization and optimal control. Problem sizes range from single-variable unconstrained minimization to constrained problems with highly nonlinear functions and hundreds of variables. Most of the applications can be posed as nonlinearly constrained minimization problems. Highly complex optimization problems with many variables were formulated in the early days of computing. At the time, many problems had to be reformulated or bypassed entirely, and solution methods often relied on problem-specific strategies. Problems with more than ten variables usually went unsolved.
Convex relaxations for gas expansion planning
Borraz-Sanchez, Conrado; Bent, Russell Whitford; Backhaus, Scott N.; Hijazi, Hassan; Van Hentenryck, Pascal
2016-01-01
Expansion of natural gas networks is a critical process involving substantial capital expenditures with complex decision-support requirements. Here, given the non-convex nature of gas transmission constraints, global optimality and infeasibility guarantees can only be offered by global optimisation approaches. Unfortunately, state-of-the-art global optimisation solvers are unable to scale up to real-world size instances. In this study, we present a convex mixed-integer second-order cone relaxation for the gas expansion planning problem under steady-state conditions. The underlying model offers tight lower bounds with high computational efficiency. In addition, the optimal solution of the relaxation can often be used to derive high-quality solutions to the original problem, leading to provably tight optimality gaps and, in some cases, global optimal solutions. The convex relaxation is based on a few key ideas, including the introduction of flux direction variables, exact McCormick relaxations, on/off constraints, and integer cuts. Numerical experiments are conducted on the traditional Belgian gas network, as well as other real larger networks. The results demonstrate both the accuracy and computational speed of the relaxation and its ability to produce high-quality solution
Convex relaxations for gas expansion planning
Borraz-Sanchez, Conrado; Bent, Russell Whitford; Backhaus, Scott N.; Hijazi, Hassan; Van Hentenryck, Pascal
2016-01-01
Expansion of natural gas networks is a critical process involving substantial capital expenditures with complex decision-support requirements. Here, given the non-convex nature of gas transmission constraints, global optimality and infeasibility guarantees can only be offered by global optimisation approaches. Unfortunately, state-of-the-art global optimisation solvers are unable to scale up to real-world size instances. In this study, we present a convex mixed-integer second-order cone relaxation for the gas expansion planning problem under steady-state conditions. The underlying model offers tight lower bounds with high computational efficiency. In addition, the optimal solution of the relaxation can often be used to derive high-quality solutionsmore » to the original problem, leading to provably tight optimality gaps and, in some cases, global optimal solutions. The convex relaxation is based on a few key ideas, including the introduction of flux direction variables, exact McCormick relaxations, on/off constraints, and integer cuts. Numerical experiments are conducted on the traditional Belgian gas network, as well as other real larger networks. The results demonstrate both the accuracy and computational speed of the relaxation and its ability to produce high-quality solution« less
Optimal control in a macroeconomic problem
NASA Astrophysics Data System (ADS)
Bulgakov, V. K.; Shatov, G. L.
2007-08-01
The Pontryagin maximum principle is used to develop an original algorithm for finding an optimal control in a macroeconomic problem. Numerical results are presented for the optimal control and optimal trajectory of the development of a regional economic system. For an optimal control satisfying a certain constraint, an invariant of a macroeconomic system is derived.
First and Second Order Necessary Conditions for Stochastic Optimal Control Problems
Bonnans, J. Frederic; Silva, Francisco J.
2012-06-15
In this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state.
Solving ptychography with a convex relaxation
Chen, Richard Y; Ou, Xiaoze; Ames, Brendan; Tropp, Joel A; Yang, Changhuei
2015-01-01
Ptychography is a powerful computational imaging technique that transforms a collection of low-resolution images into a high-resolution sample reconstruction. Unfortunately, algorithms that currently solve this reconstruction problem lack stability, robustness, and theoretical guarantees. Recently, convex optimization algorithms have improved the accuracy and reliability of several related reconstruction efforts. This paper proposes a convex formulation of the ptychography problem. This formulation has no local minima, it can be solved using a wide range of algorithms, it can incorporate appropriate noise models, and it can include multiple a priori constraints. The paper considers a specific algorithm, based on low-rank factorization, whose runtime and memory usage are near-linear in the size of the output image. Experiments demonstrate that this approach offers a 25% lower background variance on average than alternating projections, the ptychographic reconstruction algorithm that is currently in widespread use. PMID:26146480
Social Emotional Optimization Algorithm for Nonlinear Constrained Optimization Problems
NASA Astrophysics Data System (ADS)
Xu, Yuechun; Cui, Zhihua; Zeng, Jianchao
Nonlinear programming problem is one important branch in operational research, and has been successfully applied to various real-life problems. In this paper, a new approach called Social emotional optimization algorithm (SEOA) is used to solve this problem which is a new swarm intelligent technique by simulating the human behavior guided by emotion. Simulation results show that the social emotional optimization algorithm proposed in this paper is effective and efficiency for the nonlinear constrained programming problems.
Constrained Graph Optimization: Interdiction and Preservation Problems
Schild, Aaron V
2012-07-30
The maximum flow, shortest path, and maximum matching problems are a set of basic graph problems that are critical in theoretical computer science and applications. Constrained graph optimization, a variation of these basic graph problems involving modification of the underlying graph, is equally important but sometimes significantly harder. In particular, one can explore these optimization problems with additional cost constraints. In the preservation case, the optimizer has a budget to preserve vertices or edges of a graph, preventing them from being deleted. The optimizer wants to find the best set of preserved edges/vertices in which the cost constraints are satisfied and the basic graph problems are optimized. For example, in shortest path preservation, the optimizer wants to find a set of edges/vertices within which the shortest path between two predetermined points is smallest. In interdiction problems, one deletes vertices or edges from the graph with a particular cost in order to impede the basic graph problems as much as possible (for example, delete edges/vertices to maximize the shortest path between two predetermined vertices). Applications of preservation problems include optimal road maintenance, power grid maintenance, and job scheduling, while interdiction problems are related to drug trafficking prevention, network stability assessment, and counterterrorism. Computational hardness results are presented, along with heuristic methods for approximating solutions to the matching interdiction problem. Also, efficient algorithms are presented for special cases of graphs, including on planar graphs. The graphs in many of the listed applications are planar, so these algorithms have important practical implications.
A Mathematical Optimization Problem in Bioinformatics
ERIC Educational Resources Information Center
Heyer, Laurie J.
2008-01-01
This article describes the sequence alignment problem in bioinformatics. Through examples, we formulate sequence alignment as an optimization problem and show how to compute the optimal alignment with dynamic programming. The examples and sample exercises have been used by the author in a specialized course in bioinformatics, but could be adapted…
Problem Solving through an Optimization Problem in Geometry
ERIC Educational Resources Information Center
Poon, Kin Keung; Wong, Hang-Chi
2011-01-01
This article adapts the problem-solving model developed by Polya to investigate and give an innovative approach to discuss and solve an optimization problem in geometry: the Regiomontanus Problem and its application to football. Various mathematical tools, such as calculus, inequality and the properties of circles, are used to explore and reflect…
Particle swarm optimization for complex nonlinear optimization problems
NASA Astrophysics Data System (ADS)
Alexandridis, Alex; Famelis, Ioannis Th.; Tsitouras, Charalambos
2016-06-01
This work presents the application of a technique belonging to evolutionary computation, namely particle swarm optimization (PSO), to complex nonlinear optimization problems. To be more specific, a PSO optimizer is setup and applied to the derivation of Runge-Kutta pairs for the numerical solution of initial value problems. The effect of critical PSO operational parameters on the performance of the proposed scheme is thoroughly investigated.
Some shape optimization problems for eigenvalues
NASA Astrophysics Data System (ADS)
Gasimov, Yusif S.
2008-02-01
In this work we consider some inverse problems with respect to domain for the Laplace operator. The considered problems are reduced to the variational formulation. The equivalency of these problems is obtained under some conditions. The formula is obtained for the eigenvalue in the optimal domain.
Entanglement Quantification Made Easy: Polynomial Measures Invariant under Convex Decomposition
NASA Astrophysics Data System (ADS)
Regula, Bartosz; Adesso, Gerardo
2016-02-01
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
Analog Processor To Solve Optimization Problems
NASA Technical Reports Server (NTRS)
Duong, Tuan A.; Eberhardt, Silvio P.; Thakoor, Anil P.
1993-01-01
Proposed analog processor solves "traveling-salesman" problem, considered paradigm of global-optimization problems involving routing or allocation of resources. Includes electronic neural network and auxiliary circuitry based partly on concepts described in "Neural-Network Processor Would Allocate Resources" (NPO-17781) and "Neural Network Solves 'Traveling-Salesman' Problem" (NPO-17807). Processor based on highly parallel computing solves problem in significantly less time.
Dynamic programming in applied optimization problems
NASA Astrophysics Data System (ADS)
Zavalishchin, Dmitry
2015-11-01
Features of the use dynamic programming in applied problems are investigated. In practice such problems as finding the critical paths in network planning and control, finding the optimal supply plan in transportation problem, objects territorial distribution are traditionally solved by special methods of operations research. It should be noted that the dynamic programming is not provided computational advantages, but facilitates changes and modifications of tasks. This follows from the Bellman's optimality principle. The features of the multistage decision processes construction in applied problems are provided.
Exploiting phase transitions for fusion optimization problems
NASA Astrophysics Data System (ADS)
Svenson, Pontus
2005-05-01
Many optimization problems that arise in multi-target tracking and fusion applications are known to be NP-complete, ie, believed to have worst-case complexities that are exponential in problem size. Recently, many such NP-complete problems have been shown to display threshold phenomena: it is possible to define a parameter such that the probability of a random problem instance having a solution jumps from 1 to 0 at a specific value of the parameter. It is also found that the amount of resources needed to solve the problem instance peaks at the transition point. Among the problems found to display this behavior are graph coloring (aka clustering, relevant for multi-target tracking), satisfiability (which occurs in resource allocation and planning problem), and the travelling salesperson problem. Physicists studying these problems have found intriguing similarities to phase transitions in spin models of statistical mechanics. Many methods previously used to analyze spin glasses have been used to explain some of the properties of the behavior at the transition point. It turns out that the transition happens because the fitness landscape of the problem changes as the parameter is varied. Some algorithms have been introduced that exploit this knowledge of the structure of the fitness landscape. In this paper, we review some of the experimental and theoretical work on threshold phenomena in optimization problems and indicate how optimization problems from tracking and sensor resource allocation could be analyzed using these results.
A Problem-Oriented Mathematical Optimization Course
ERIC Educational Resources Information Center
Wenger, Robert B.; Rhyner, Charles R.
1977-01-01
This article describes a one-semester junior and senior level course in applied mathematical optimization for undergraduates which provides the opportunity to test models by doing numerical experiments and to learn to use computer subroutines for solving optimization problems. (MN)
Random generation of structured linear optimization problems
Arthur, J.; Frendewey, J. Jr.
1994-12-31
We describe the on-going development of a random generator for linear optimization problems (LPs) founded on the concept of block structure. The general LP: minimize z = cx subject to Ax = b, x {ge} 0 can take a variety of special forms determined (primarily) by predefined structures on the matrix A of constraint coefficients. The authors have developed several random problem generators which provide instances of LPs having such structure; in particular (i) general (non-structured) problems, (ii) generalized upper bound (GUB) constraints, (iii) minimum cost network flow problems, (iv) transportation and assignment problems, (v) shortest path problems, (vi) generalized network flow problems, and (vii) multicommodity network flow problems. This paper discusses the general philosophy behind the construction of these generators. In addition, the task of combining the generators into a single generator -- in which the matrix A can contain various blocks, each of a prescribed structure from those mentioned above -- is described.
Solving constrained optimization problems with hybrid particle swarm optimization
NASA Astrophysics Data System (ADS)
Zahara, Erwie; Hu, Chia-Hsin
2008-11-01
Constrained optimization problems (COPs) are very important in that they frequently appear in the real world. A COP, in which both the function and constraints may be nonlinear, consists of the optimization of a function subject to constraints. Constraint handling is one of the major concerns when solving COPs with particle swarm optimization (PSO) combined with the Nelder-Mead simplex search method (NM-PSO). This article proposes embedded constraint handling methods, which include the gradient repair method and constraint fitness priority-based ranking method, as a special operator in NM-PSO for dealing with constraints. Experiments using 13 benchmark problems are explained and the NM-PSO results are compared with the best known solutions reported in the literature. Comparison with three different meta-heuristics demonstrates that NM-PSO with the embedded constraint operator is extremely effective and efficient at locating optimal solutions.
Graph optimization problems on a Bethe lattice
NASA Astrophysics Data System (ADS)
de Oliveira, Mário J.
1989-01-01
The p-partitioning and p-coloring problems on a Bethe lattice of coordination number z are analyzed. It is shown that these two NP-complete optimization problems turn out to be equivalent to finding the ground-state energy of p-state Potts models with frustration. Numerical calculation of the cost function of both problems are carried out for several values of z and p. In the case of p=2 the results are identical to those obtained by Mézard and Parisi for the case of the bipartitioning problem. A numerical upper bound to the chromatic number is found for several values of z.
Quantum optimization and maximum clique problems
NASA Astrophysics Data System (ADS)
Yatsenko, Vitaliy A.; Pardalos, Panos M.; Chiarini, Bruno H.
2004-08-01
This paper describes a new approach to global optimization and control uses geometric methods and modern quantum mathematics. Polynomial extremal problems (PEP) are considered. PEP constitute one of the most important subclasses of nonlinear programming models. Their distinctive feature is that an objective function and constraints can be expressed by polynomial functions in one or several variables. A general approach to optimization based on quantum holonomic computing algorithms and instanton mechanism. An optimization method based on geometric Lie - algebraic structures on Grassmann manifolds and related with Lax type flows is proposed. Making use of the differential geometric techniques it is shown that associated holonomy groups properly realizing quantum computation can be effectively found concerning polynomial problems. Two examples demonstrating calculation aspects of holonomic quantum computer and maximum clique problems in very large graphs, are considered in detail.
Stochastic time-optimal control problems
NASA Technical Reports Server (NTRS)
Zhang, W.; Elliot, D.
1988-01-01
Two types of stochastic time-optimal controls in a one-dimensional setting are considered. Multidimensional problems, in the case of complete state information available and the system modeled by stochastic differential equations, are studied under the formulation of minimizing the expected transient-response time. The necessary condition of optimality is the satisfaction for the value function of a parabolic partial differential equation with boundary conditions. The sufficient condition of optimality is also provided, based on Dynkin's formula. Finally, three examples are given.
The inverse problem of the optimal regulator.
NASA Technical Reports Server (NTRS)
Yokoyama, R.; Kinnen, E.
1972-01-01
The inverse problem of the optimal regulator is considered for a general class of multi-input systems with integral-type performance indices. A new phase variable canonical form is shown to be convenient for this analysis. The advantage of the canonical form is to separate the state variables into subvectors of directly controlled, indirectly controlled, and uncontrollable components. Necessary and sufficient conditions for optimized performance indices are given. With the nonlinearities of the system restricted to functions of the directly controlled state variables, additional results are developed about the nonnegative property of optimized loss functions.
Problem size, parallel architecture, and optimal speedup
NASA Technical Reports Server (NTRS)
Nicol, David M.; Willard, Frank H.
1988-01-01
The communication and synchronization overhead inherent in parallel processing can lead to situations where adding processors to the solution method actually increases execution time. Problem type, problem size, and architecture type all affect the optimal number of processors to employ. The numerical solution of an elliptic partial differential equation is examined in order to study the relationship between problem size and architecture. The equation's domain is discretized into n sup 2 grid points which are divided into partitions and mapped onto the individual processor memories. The relationships between grid size, stencil type, partitioning strategy, processor execution time, and communication network type are analytically quantified. In so doing, the optimal number of processors was determined to assign to the solution, and identified (1) the smallest grid size which fully benefits from using all available processors, (2) the leverage on performance given by increasing processor speed or communication network speed, and (3) the suitability of various architectures for large numerical problems.
Problem size, parallel architecture and optimal speedup
NASA Technical Reports Server (NTRS)
Nicol, David M.; Willard, Frank H.
1987-01-01
The communication and synchronization overhead inherent in parallel processing can lead to situations where adding processors to the solution method actually increases execution time. Problem type, problem size, and architecture type all affect the optimal number of processors to employ. The numerical solution of an elliptic partial differential equation is examined in order to study the relationship between problem size and architecture. The equation's domain is discretized into n sup 2 grid points which are divided into partitions and mapped onto the individual processor memories. The relationships between grid size, stencil type, partitioning strategy, processor execution time, and communication network type are analytically quantified. In so doing, the optimal number of processors was determined to assign to the solution, and identified (1) the smallest grid size which fully benefits from using all available processors, (2) the leverage on performance given by increasing processor speed or communication network speed, and (3) the suitability of various architectures for large numerical problems.
Linear stochastic optimal control and estimation problem
NASA Technical Reports Server (NTRS)
Geyser, L. C.; Lehtinen, F. K. B.
1980-01-01
Problem involves design of controls for linear time-invariant system disturbed by white noise. Solution is Kalman filter coupled through set of optimal regulator gains to produce desired control signal. Key to solution is solving matrix Riccati differential equation. LSOCE effectively solves problem for wide range of practical applications. Program is written in FORTRAN IV for batch execution and has been implemented on IBM 360.
Solving global optimization problems on GPU cluster
NASA Astrophysics Data System (ADS)
Barkalov, Konstantin; Gergel, Victor; Lebedev, Ilya
2016-06-01
The paper contains the results of investigation of a parallel global optimization algorithm combined with a dimension reduction scheme. This allows solving multidimensional problems by means of reducing to data-independent subproblems with smaller dimension solved in parallel. The new element implemented in the research consists in using several graphic accelerators at different computing nodes. The paper also includes results of solving problems of well-known multiextremal test class GKLS on Lobachevsky supercomputer using tens of thousands of GPU cores.
Maximum margin classification based on flexible convex hulls for fault diagnosis of roller bearings
NASA Astrophysics Data System (ADS)
Zeng, Ming; Yang, Yu; Zheng, Jinde; Cheng, Junsheng
2016-01-01
A maximum margin classification based on flexible convex hulls (MMC-FCH) is proposed and applied to fault diagnosis of roller bearings. In this method, the class region of each sample set is approximated by a flexible convex hull of its training samples, and then an optimal separating hyper-plane that maximizes the geometric margin between flexible convex hulls is constructed by solving a closest pair of points problem. By using the kernel trick, MMC-FCH can be extended to nonlinear cases. In addition, multi-class classification problems can be processed by constructing binary pairwise classifiers as in support vector machine (SVM). Actually, the classical SVM also can be regarded as a maximum margin classification based on convex hulls (MMC-CH), which approximates each class region with a convex hull. The convex hull is a special case of the flexible convex hull. To train a MMC-FCH classifier, time-domain and frequency-domain statistical parameters are extracted not only from raw vibration signals but also from the resulting intrinsic mode functions (IMFs) by performing empirical mode decomposition (EMD) on the raw signals, and then the distance evaluation technique (DET) is used to select salient features from the whole statistical features. The experiments on bearing datasets show that the proposed method can reliably recognize different bearing faults.
NASA Astrophysics Data System (ADS)
Skala, Vaclav
2016-06-01
There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees, octrees, kd-trees, bounding volume hierarchies etc. However in some applications a non-orthogonal space subdivision can offer new ways for actual speed up. In the case of convex polygon in E2 a simple Point-in-Polygon test is of the O(N) complexity and the optimal algorithm is of O(log N) computational complexity. In the E3 case, the complexity is O(N) even for the convex polyhedron as no ordering is defined. New Point-in-Convex Polygon and Point-in-Convex Polyhedron algorithms are presented based on space subdivision in the preprocessing stage resulting to O(1) run-time complexity. The presented approach is simple to implement. Due to the principle of duality, dual problems, e.g. line-convex polygon, line clipping, can be solved in a similarly.
First-order convex feasibility algorithms for x-ray CT
Sidky, Emil Y.; Pan Xiaochuan; Jorgensen, Jakob S.
2013-03-15
Purpose: Iterative image reconstruction (IIR) algorithms in computed tomography (CT) are based on algorithms for solving a particular optimization problem. Design of the IIR algorithm, therefore, is aided by knowledge of the solution to the optimization problem on which it is based. Often times, however, it is impractical to achieve accurate solution to the optimization of interest, which complicates design of IIR algorithms. This issue is particularly acute for CT with a limited angular-range scan, which leads to poorly conditioned system matrices and difficult to solve optimization problems. In this paper, we develop IIR algorithms which solve a certain type of optimization called convex feasibility. The convex feasibility approach can provide alternatives to unconstrained optimization approaches and at the same time allow for rapidly convergent algorithms for their solution-thereby facilitating the IIR algorithm design process. Methods: An accelerated version of the Chambolle-Pock (CP) algorithm is adapted to various convex feasibility problems of potential interest to IIR in CT. One of the proposed problems is seen to be equivalent to least-squares minimization, and two other problems provide alternatives to penalized, least-squares minimization. Results: The accelerated CP algorithms are demonstrated on a simulation of circular fan-beam CT with a limited scanning arc of 144 Degree-Sign . The CP algorithms are seen in the empirical results to converge to the solution of their respective convex feasibility problems. Conclusions: Formulation of convex feasibility problems can provide a useful alternative to unconstrained optimization when designing IIR algorithms for CT. The approach is amenable to recent methods for accelerating first-order algorithms which may be particularly useful for CT with limited angular-range scanning. The present paper demonstrates the methodology, and future work will illustrate its utility in actual CT application.
First-order convex feasibility algorithms for x-ray CT
Sidky, Emil Y.; Jørgensen, Jakob S.; Pan, Xiaochuan
2013-01-01
Purpose: Iterative image reconstruction (IIR) algorithms in computed tomography (CT) are based on algorithms for solving a particular optimization problem. Design of the IIR algorithm, therefore, is aided by knowledge of the solution to the optimization problem on which it is based. Often times, however, it is impractical to achieve accurate solution to the optimization of interest, which complicates design of IIR algorithms. This issue is particularly acute for CT with a limited angular-range scan, which leads to poorly conditioned system matrices and difficult to solve optimization problems. In this paper, we develop IIR algorithms which solve a certain type of optimization called convex feasibility. The convex feasibility approach can provide alternatives to unconstrained optimization approaches and at the same time allow for rapidly convergent algorithms for their solution—thereby facilitating the IIR algorithm design process. Methods: An accelerated version of the Chambolle−Pock (CP) algorithm is adapted to various convex feasibility problems of potential interest to IIR in CT. One of the proposed problems is seen to be equivalent to least-squares minimization, and two other problems provide alternatives to penalized, least-squares minimization. Results: The accelerated CP algorithms are demonstrated on a simulation of circular fan-beam CT with a limited scanning arc of 144°. The CP algorithms are seen in the empirical results to converge to the solution of their respective convex feasibility problems. Conclusions: Formulation of convex feasibility problems can provide a useful alternative to unconstrained optimization when designing IIR algorithms for CT. The approach is amenable to recent methods for accelerating first-order algorithms which may be particularly useful for CT with limited angular-range scanning. The present paper demonstrates the methodology, and future work will illustrate its utility in actual CT application. PMID:23464295
Solving optimization problems on computational grids.
Wright, S. J.; Mathematics and Computer Science
2001-05-01
Multiprocessor computing platforms, which have become more and more widely available since the mid-1980s, are now heavily used by organizations that need to solve very demanding computational problems. Parallel computing is now central to the culture of many research communities. Novel parallel approaches were developed for global optimization, network optimization, and direct-search methods for nonlinear optimization. Activity was particularly widespread in parallel branch-and-bound approaches for various problems in combinatorial and network optimization. As the cost of personal computers and low-end workstations has continued to fall, while the speed and capacity of processors and networks have increased dramatically, 'cluster' platforms have become popular in many settings. A somewhat different type of parallel computing platform know as a computational grid (alternatively, metacomputer) has arisen in comparatively recent times. Broadly speaking, this term refers not to a multiprocessor with identical processing nodes but rather to a heterogeneous collection of devices that are widely distributed, possibly around the globe. The advantage of such platforms is obvious: they have the potential to deliver enormous computing power. Just as obviously, however, the complexity of grids makes them very difficult to use. The Condor team, headed by Miron Livny at the University of Wisconsin, were among the pioneers in providing infrastructure for grid computations. More recently, the Globus project has developed technologies to support computations on geographically distributed platforms consisting of high-end computers, storage and visualization devices, and other scientific instruments. In 1997, we started the metaneos project as a collaborative effort between optimization specialists and the Condor and Globus groups. Our aim was to address complex, difficult optimization problems in several areas, designing and implementing the algorithms and the software
Statistical Physics of Hard Optimization Problems
NASA Astrophysics Data System (ADS)
Zdeborová, Lenka
2008-06-01
Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the NP-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this thesis is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.
Statistical physics of hard optimization problems
NASA Astrophysics Data System (ADS)
Zdeborová, Lenka
2009-06-01
Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.
Evolutionary strategies for solving optimization problems
NASA Astrophysics Data System (ADS)
Ebeling, Werner; Reimann, Axel; Molgedey, Lutz
We will give a survey of applications of thermodynamically and biologically oriented evolutionary strategies for optimization problems. Primarily, we investigate the solution of discrete optimization problems, most of combinatorial type, using a certain class of coupled differential equations. The problem is to find the minimum on a large set of real numbers (the potential) Ui, defined on the integer set i = 1 ...s, where s is an extremely large nu mber. The stationary states of the system correspond to relative optima on the discrete set. First, several elementary evolutionary strategies are described by simple deterministic equations, leading to a high-dimensional system of coupled differential equations. The known equations for thermodynamic search processes and for simple models of biological evolution are unified by defining a two-parameter family of equations which embed both cases. The unified equations model mixed Boltzmann/Darwin- strategies including basic elements of thermodynamical and biological evolution as well. In a next step a master equation model in the occupation number space is defined. We investigate the transition probabilities and the convergence properties using tools from the theory of stochastic processes. Several examples are analyzed. In particular we study the optimization of theoretical model sequences with simple valuation rules. In order to demonstrate that the strategies developed here may also be used to investigate realistic problems we present an example application to RNA folding (search for a minimum free energy configuration).
Optimal solutions of unobservable orbit determination problems
NASA Astrophysics Data System (ADS)
Cicci, David A.; Tapley, Byron D.
1988-12-01
The method of data augmentation, in the form ofa priori covariance information on the reference solution, as a means to overcome the effects of ill-conditioning in orbit determination problems has been investigated. Specifically, for the case when ill-conditioning results from parameter non-observability and an appropriatea priori covariance is unknown, methods by which thea priori covariance is optimally chosen are presented. In problems where an inaccuratea priori covariance is provided, the optimal weighting of this data set is obtained. The feasibility of these ‘ridge-type’ solution methods is demonstrated by their application to a non-observable gravity field recovery simulation. In the simulation, both ‘ridge-type’ and conventional solutions are compared. Substantial improvement in the accuracy of the conventional solution is realized by the use of these ridge-type solution methods. The solution techniques presented in this study are applicable to observable, but ill-conditioned problems as well as the unobservable problems directly addressed. For the case of observable problems, the ridge-type solutions provide an improvement in the accuracy of the ordinary least squares solutions.
Hierarchical optimization for neutron scattering problems
NASA Astrophysics Data System (ADS)
Bao, Feng; Archibald, Rick; Bansal, Dipanshu; Delaire, Olivier
2016-06-01
We present a scalable optimization method for neutron scattering problems that determines confidence regions of simulation parameters in lattice dynamics models used to fit neutron scattering data for crystalline solids. The method uses physics-based hierarchical dimension reduction in both the computational simulation domain and the parameter space. We demonstrate for silicon that after a few iterations the method converges to parameters values (interatomic force-constants) computed with density functional theory simulations.
Interaction Prediction Optimization in Multidisciplinary Design Optimization Problems
Zhang, Xiaoling; Huang, Hong-Zhong; Wang, Zhonglai; Xu, Huanwei
2014-01-01
The distributed strategy of Collaborative Optimization (CO) is suitable for large-scale engineering systems. However, it is hard for CO to converge when there is a high level coupled dimension. Furthermore, the discipline objectives cannot be considered in each discipline optimization problem. In this paper, one large-scale systems control strategy, the interaction prediction method (IPM), is introduced to enhance CO. IPM is utilized for controlling subsystems and coordinating the produce process in large-scale systems originally. We combine the strategy of IPM with CO and propose the Interaction Prediction Optimization (IPO) method to solve MDO problems. As a hierarchical strategy, there are a system level and a subsystem level in IPO. The interaction design variables (including shared design variables and linking design variables) are operated at the system level and assigned to the subsystem level as design parameters. Each discipline objective is considered and optimized at the subsystem level simultaneously. The values of design variables are transported between system level and subsystem level. The compatibility constraints are replaced with the enhanced compatibility constraints to reduce the dimension of design variables in compatibility constraints. Two examples are presented to show the potential application of IPO for MDO. PMID:24744685
A novel metaheuristic for continuous optimization problems: Virus optimization algorithm
NASA Astrophysics Data System (ADS)
Liang, Yun-Chia; Rodolfo Cuevas Juarez, Josue
2016-01-01
A novel metaheuristic for continuous optimization problems, named the virus optimization algorithm (VOA), is introduced and investigated. VOA is an iteratively population-based method that imitates the behaviour of viruses attacking a living cell. The number of viruses grows at each replication and is controlled by an immune system (a so-called 'antivirus') to prevent the explosive growth of the virus population. The viruses are divided into two classes (strong and common) to balance the exploitation and exploration effects. The performance of the VOA is validated through a set of eight benchmark functions, which are also subject to rotation and shifting effects to test its robustness. Extensive comparisons were conducted with over 40 well-known metaheuristic algorithms and their variations, such as artificial bee colony, artificial immune system, differential evolution, evolutionary programming, evolutionary strategy, genetic algorithm, harmony search, invasive weed optimization, memetic algorithm, particle swarm optimization and simulated annealing. The results showed that the VOA is a viable solution for continuous optimization.
On domains of convergence in optimization problems
NASA Technical Reports Server (NTRS)
Diaz, Alejandro R.; Shaw, Steven S.; Pan, Jian
1990-01-01
Numerical optimization algorithms require the knowledge of an initial set of design variables. Starting from an initial design x(sup 0), improved solutions are obtained by updating the design iteratively in a way prescribed by the particular algorithm used. If the algorithm is successful, convergence is achieved to a local optimal solution. Let A denote the iterative procedure that characterizes a typical optimization algorithm, applied to the problem: Find x belonging to R(sup n) that maximizes f(x) subject to x belonging to Omega contained in R(sup n). We are interested in problems with several local maxima (x(sub j))(sup *), j=1, ..., m, in the feasible design space Omega. In general, convergence of the algorithm A to a specific solution (x(sub j))(sup *) is determined by the choice of initial design x(sup 0). The domain of convergence D(sub j) of A associated with a local maximum (x(sub j))(sup *) is a subset of initial designs x(sup 0) in Omega such that the sequence (x(sup k)), k=0,1,2,... defined by x(sup k+1) = A(x(sup k)), k=0,1,... converges to (x(sub j))(sup *). The set D(sub j) is also called the basin of attraction of (x(sub j))(sup *). Cayley first proposed the problem of finding the basin of attraction for Newton's method in 1897. It has been shown that the basin of attraction for Newton's method exhibits chaotic behavior in problems with polynomial objective. This implies that there may be regions in the feasible design space where arbitrarily close starting points will converge to different local optimal solutions. Furthermore, the boundaries of the domains of convergence may have a very complex, even fractal structure. In this paper we show that even simple structural optimization problems solved using standard gradient based (first order) algorithms exhibit similar features.
NASA Technical Reports Server (NTRS)
Tennille, Geoffrey M.; Howser, Lona M.
1993-01-01
The use of the CONVEX computers that are an integral part of the Supercomputing Network Subsystems (SNS) of the Central Scientific Computing Complex of LaRC is briefly described. Features of the CONVEX computers that are significantly different than the CRAY supercomputers are covered, including: FORTRAN, C, architecture of the CONVEX computers, the CONVEX environment, batch job submittal, debugging, performance analysis, utilities unique to CONVEX, and documentation. This revision reflects the addition of the Applications Compiler and X-based debugger, CXdb. The document id intended for all CONVEX users as a ready reference to frequently asked questions and to more detailed information contained with the vendor manuals. It is appropriate for both the novice and the experienced user.
Optimal pre-scheduling of problem remappings
NASA Technical Reports Server (NTRS)
Nicol, David M.; Saltz, Joel H.
1987-01-01
A large class of scientific computational problems can be characterized as a sequence of steps where a significant amount of computation occurs each step, but the work performed at each step is not necessarily identical. Two good examples of this type of computation are: (1) regridding methods which change the problem discretization during the course of the computation, and (2) methods for solving sparse triangular systems of linear equations. Recent work has investigated a means of mapping such computations onto parallel processors; the method defines a family of static mappings with differing degrees of importance placed on the conflicting goals of good load balance and low communication/synchronization overhead. The performance tradeoffs are controllable by adjusting the parameters of the mapping method. To achieve good performance it may be necessary to dynamically change these parameters at run-time, but such changes can impose additional costs. If the computation's behavior can be determined prior to its execution, it can be possible to construct an optimal parameter schedule using a low-order-polynomial-time dynamic programming algorithm. Since the latter can be expensive, the performance is studied of the effect of a linear-time scheduling heuristic on one of the model problems, and it is shown to be effective and nearly optimal.
Optimal Planning and Problem-Solving
NASA Technical Reports Server (NTRS)
Clemet, Bradley; Schaffer, Steven; Rabideau, Gregg
2008-01-01
CTAEMS MDP Optimal Planner is a problem-solving software designed to command a single spacecraft/rover, or a team of spacecraft/rovers, to perform the best action possible at all times according to an abstract model of the spacecraft/rover and its environment. It also may be useful in solving logistical problems encountered in commercial applications such as shipping and manufacturing. The planner reasons around uncertainty according to specified probabilities of outcomes using a plan hierarchy to avoid exploring certain kinds of suboptimal actions. Also, planned actions are calculated as the state-action space is expanded, rather than afterward, to reduce by an order of magnitude the processing time and memory used. The software solves planning problems with actions that can execute concurrently, that have uncertain duration and quality, and that have functional dependencies on others that affect quality. These problems are modeled in a hierarchical planning language called C_TAEMS, a derivative of the TAEMS language for specifying domains for the DARPA Coordinators program. In realistic environments, actions often have uncertain outcomes and can have complex relationships with other tasks. The planner approaches problems by considering all possible actions that may be taken from any state reachable from a given, initial state, and from within the constraints of a given task hierarchy that specifies what tasks may be performed by which team member.
The Replica Method in Optimization Problems.
NASA Astrophysics Data System (ADS)
Liao, Wuwell W.
In this thesis I discuss the application of the replica method in combinatorial optimization problems. In particular, I study certain graph-partitioning problems. One problem that I consider is the following. We are given a set of vertices V = (V_1,V_2,ldots V_{N}), with N even, and a set of edges E = {(V_{i},V _{j})i not= j}. Let each edge be connected with probability P. The bipartitioning problem is to divide V into two parts of equal size, in such a way as to minimize the number of edges N _{c} connecting these two parts. We are interested in the behavior of N_{c }/N, averaged over all possible configurations of edges in the limit N --> infty , as a function of the connectivity alpha = NP. When alpha is finite, the problem is shown to be similar, but not identical, to the mean field theory of a spin glass with finite connectivity. The replica-symmetric solution is derived. It is shown to be consistent with exact results for the infinite cluster obtained by P. Erdos.
Hybrid intelligent optimization methods for engineering problems
NASA Astrophysics Data System (ADS)
Pehlivanoglu, Yasin Volkan
The purpose of optimization is to obtain the best solution under certain conditions. There are numerous optimization methods because different problems need different solution methodologies; therefore, it is difficult to construct patterns. Also mathematical modeling of a natural phenomenon is almost based on differentials. Differential equations are constructed with relative increments among the factors related to yield. Therefore, the gradients of these increments are essential to search the yield space. However, the landscape of yield is not a simple one and mostly multi-modal. Another issue is differentiability. Engineering design problems are usually nonlinear and they sometimes exhibit discontinuous derivatives for the objective and constraint functions. Due to these difficulties, non-gradient-based algorithms have become more popular in recent decades. Genetic algorithms (GA) and particle swarm optimization (PSO) algorithms are popular, non-gradient based algorithms. Both are population-based search algorithms and have multiple points for initiation. A significant difference from a gradient-based method is the nature of the search methodologies. For example, randomness is essential for the search in GA or PSO. Hence, they are also called stochastic optimization methods. These algorithms are simple, robust, and have high fidelity. However, they suffer from similar defects, such as, premature convergence, less accuracy, or large computational time. The premature convergence is sometimes inevitable due to the lack of diversity. As the generations of particles or individuals in the population evolve, they may lose their diversity and become similar to each other. To overcome this issue, we studied the diversity concept in GA and PSO algorithms. Diversity is essential for a healthy search, and mutations are the basic operators to provide the necessary variety within a population. After having a close scrutiny of the diversity concept based on qualification and
Recognition of Graphs with Convex Quadratic Stability Number
NASA Astrophysics Data System (ADS)
Pacheco, Maria F.; Cardoso, Domingos M.
2009-09-01
A stable set of a graph is a set of mutually non-adjacent vertices. The determination of a maximum size stable set, which is called maximum stable set, and the determination of its size, which is called stability number, are central combinatorial optimization problems. However, given a nonnegative integer k, to determine if a graph G has a stable set of size k is NP-complete. In this paper we deal with graphs for which the stability number can be determined by solving a convex quadratic programming problem. Such graphs were introduced in [13] and are called graphs with convex-QP stability number. A few algorithmic techniques for the recognition of this type of graphs in particular families are presented.
ERIC Educational Resources Information Center
Hodge, Jonathan K.; Marshall, Emily; Patterson, Geoff
2010-01-01
Convexity-based measures of shape compactness provide an effective way to identify irregularities in congressional district boundaries. A low convexity coefficient may suggest that a district has been gerrymandered, or it may simply reflect irregularities in the corresponding state boundary. Furthermore, the distribution of population within a…
LDRD Final Report: Global Optimization for Engineering Science Problems
HART,WILLIAM E.
1999-12-01
For a wide variety of scientific and engineering problems the desired solution corresponds to an optimal set of objective function parameters, where the objective function measures a solution's quality. The main goal of the LDRD ''Global Optimization for Engineering Science Problems'' was the development of new robust and efficient optimization algorithms that can be used to find globally optimal solutions to complex optimization problems. This SAND report summarizes the technical accomplishments of this LDRD, discusses lessons learned and describes open research issues.
Tunneling and Speedup in Quantum Optimization for Permutation-Symmetric Problems
NASA Astrophysics Data System (ADS)
Muthukrishnan, Siddharth; Albash, Tameem; Lidar, Daniel A.
2016-07-01
Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semiclassical potential arising from the spin-coherent path-integral formalism. We then provide an example where the shape of the barrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoided-level crossings, involving no tunneling, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spin-vector dynamics, is at least as efficient as diabatic QA. Finally, in a different example with a convex cost function, the diabatic transitions result in a speedup relative to both adiabatic QA with tunneling and classical spin-vector dynamics.
Convex-relaxed kernel mapping for image segmentation.
Ben Salah, Mohamed; Ben Ayed, Ismail; Jing Yuan; Hong Zhang
2014-03-01
This paper investigates a convex-relaxed kernel mapping formulation of image segmentation. We optimize, under some partition constraints, a functional containing two characteristic terms: 1) a data term, which maps the observation space to a higher (possibly infinite) dimensional feature space via a kernel function, thereby evaluating nonlinear distances between the observations and segments parameters and 2) a total-variation term, which favors smooth segment surfaces (or boundaries). The algorithm iterates two steps: 1) a convex-relaxation optimization with respect to the segments by solving an equivalent constrained problem via the augmented Lagrange multiplier method and 2) a convergent fixed-point optimization with respect to the segments parameters. The proposed algorithm can bear with a variety of image types without the need for complex and application-specific statistical modeling, while having the computational benefits of convex relaxation. Our solution is amenable to parallelized implementations on graphics processing units (GPUs) and extends easily to high dimensions. We evaluated the proposed algorithm with several sets of comprehensive experiments and comparisons, including: 1) computational evaluations over 3D medical-imaging examples and high-resolution large-size color photographs, which demonstrate that a parallelized implementation of the proposed method run on a GPU can bring a significant speed-up and 2) accuracy evaluations against five state-of-the-art methods over the Berkeley color-image database and a multimodel synthetic data set, which demonstrates competitive performances of the algorithm. PMID:24723519
Convex accelerated maximum entropy reconstruction
NASA Astrophysics Data System (ADS)
Worley, Bradley
2016-04-01
Maximum entropy (MaxEnt) spectral reconstruction methods provide a powerful framework for spectral estimation of nonuniformly sampled datasets. Many methods exist within this framework, usually defined based on the magnitude of a Lagrange multiplier in the MaxEnt objective function. An algorithm is presented here that utilizes accelerated first-order convex optimization techniques to rapidly and reliably reconstruct nonuniformly sampled NMR datasets using the principle of maximum entropy. This algorithm - called CAMERA for Convex Accelerated Maximum Entropy Reconstruction Algorithm - is a new approach to spectral reconstruction that exhibits fast, tunable convergence in both constant-aim and constant-lambda modes. A high-performance, open source NMR data processing tool is described that implements CAMERA, and brief comparisons to existing reconstruction methods are made on several example spectra.
NASA Astrophysics Data System (ADS)
Cheng, Jieyu; Qiu, Wu; Yuan, Jing; Fenster, Aaron; Chiu, Bernard
2016-03-01
Registration of longitudinally acquired 3D ultrasound (US) images plays an important role in monitoring and quantifying progression/regression of carotid atherosclerosis. We introduce an image-based non-rigid registration algorithm to align the baseline 3D carotid US with longitudinal images acquired over several follow-up time points. This algorithm minimizes the sum of absolute intensity differences (SAD) under a variational optical-flow perspective within a multi-scale optimization framework to capture local and global deformations. Outer wall and lumen were segmented manually on each image, and the performance of the registration algorithm was quantified by Dice similarity coefficient (DSC) and mean absolute distance (MAD) of the outer wall and lumen surfaces after registration. In this study, images for 5 subjects were registered initially by rigid registration, followed by the proposed algorithm. Mean DSC generated by the proposed algorithm was 79:3+/-3:8% for lumen and 85:9+/-4:0% for outer wall, compared to 73:9+/-3:4% and 84:7+/-3:2% generated by rigid registration. Mean MAD of 0:46+/-0:08mm and 0:52+/-0:13mm were generated for lumen and outer wall respectively by the proposed algorithm, compared to 0:55+/-0:08mm and 0:54+/-0:11mm generated by rigid registration. The mean registration time of our method per image pair was 143+/-23s.
A convex minimization approach to data association with prior constraints
NASA Astrophysics Data System (ADS)
Chen, Huimin; Kirubarajan, Thiagalingam
2004-08-01
In this paper we propose a new formulation for reliably solving the measurement-to-track association problem with a priori constraints. Those constraints are incorporated into the scalar objective function in a general formula. This is a key step in most target tracking problems when one has to handle the measurement origin uncertainty. Our methodology is able to formulate the measurement-to-track correspondence problem with most of the commonly used assumptions and considers target feature measurements and possibly unresolved measurements as well. The resulting constrained optimization problem deals with the whole combinatorial space of possible feature selections and measurement-to-track correspondences. To find the global optimal solution, we build a convex objective function and relax the integer constraint. The special structure of this extended problem assures its equivalence to the original one, but it can be solved optimally by efficient algorithms to avoid the cominatorial search. This approach works for any cost function with continuous second derivatives. We use a track formation example and a multisensor tracking scenario to illustrate the effectiveness of the convex programming approach.
NASA Technical Reports Server (NTRS)
Nguyen, Duc T.
1990-01-01
Practical engineering application can often be formulated in the form of a constrained optimization problem. There are several solution algorithms for solving a constrained optimization problem. One approach is to convert a constrained problem into a series of unconstrained problems. Furthermore, unconstrained solution algorithms can be used as part of the constrained solution algorithms. Structural optimization is an iterative process where one starts with an initial design, a finite element structure analysis is then performed to calculate the response of the system (such as displacements, stresses, eigenvalues, etc.). Based upon the sensitivity information on the objective and constraint functions, an optimizer such as ADS or IDESIGN, can be used to find the new, improved design. For the structural analysis phase, the equation solver for the system of simultaneous, linear equations plays a key role since it is needed for either static, or eigenvalue, or dynamic analysis. For practical, large-scale structural analysis-synthesis applications, computational time can be excessively large. Thus, it is necessary to have a new structural analysis-synthesis code which employs new solution algorithms to exploit both parallel and vector capabilities offered by modern, high performance computers such as the Convex, Cray-2 and Cray-YMP computers. The objective of this research project is, therefore, to incorporate the latest development in the parallel-vector equation solver, PVSOLVE into the widely popular finite-element production code, such as the SAP-4. Furthermore, several nonlinear unconstrained optimization subroutines have also been developed and tested under a parallel computer environment. The unconstrained optimization subroutines are not only useful in their own right, but they can also be incorporated into a more popular constrained optimization code, such as ADS.
Generalized convexity and inequalities
NASA Astrophysics Data System (ADS)
Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M.
2007-11-01
Let and let be the family of all mean values of two numbers in (some examples are the arithmetic, geometric, and harmonic means). Given , we say that a function is (m1,m2)-convex if f(m1(x,y))[less-than-or-equals, slant]m2(f(x),f(y)) for all . The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1,m2)-convexity on m1 and m2 and give sufficient conditions for (m1,m2)-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function.
Stereotype locally convex spaces
NASA Astrophysics Data System (ADS)
Akbarov, S. S.
2000-08-01
We give complete proofs of some previously announced results in the theory of stereotype (that is, reflexive in the sense of Pontryagin duality) locally convex spaces. These spaces have important applications in topological algebra and functional analysis.
Approximating convex Pareto surfaces in multiobjective radiotherapy planning
Craft, David L.; Halabi, Tarek F.; Shih, Helen A.; Bortfeld, Thomas R.
2006-09-15
Radiotherapy planning involves inherent tradeoffs: the primary mission, to treat the tumor with a high, uniform dose, is in conflict with normal tissue sparing. We seek to understand these tradeoffs on a case-to-case basis, by computing for each patient a database of Pareto optimal plans. A treatment plan is Pareto optimal if there does not exist another plan which is better in every measurable dimension. The set of all such plans is called the Pareto optimal surface. This article presents an algorithm for computing well distributed points on the (convex) Pareto optimal surface of a multiobjective programming problem. The algorithm is applied to intensity-modulated radiation therapy inverse planning problems, and results of a prostate case and a skull base case are presented, in three and four dimensions, investigating tradeoffs between tumor coverage and critical organ sparing.
Group Search Optimizer for the Mobile Location Management Problem
Wang, Dan; Xiong, Congcong; Huang, Wei
2014-01-01
We propose a diversity-guided group search optimizer-based approach for solving the location management problem in mobile computing. The location management problem, which is to find the optimal network configurations of management under the mobile computing environment, is considered here as an optimization problem. The proposed diversity-guided group search optimizer algorithm is realized with the aid of diversity operator, which helps alleviate the premature convergence problem of group search optimizer algorithm, a successful optimization algorithm inspired by the animal behavior. To address the location management problem, diversity-guided group search optimizer algorithm is exploited to optimize network configurations of management by minimizing the sum of location update cost and location paging cost. Experimental results illustrate the effectiveness of the proposed approach. PMID:25180199
Group search optimizer for the mobile location management problem.
Wang, Dan; Xiong, Congcong; Huang, Wei
2014-01-01
We propose a diversity-guided group search optimizer-based approach for solving the location management problem in mobile computing. The location management problem, which is to find the optimal network configurations of management under the mobile computing environment, is considered here as an optimization problem. The proposed diversity-guided group search optimizer algorithm is realized with the aid of diversity operator, which helps alleviate the premature convergence problem of group search optimizer algorithm, a successful optimization algorithm inspired by the animal behavior. To address the location management problem, diversity-guided group search optimizer algorithm is exploited to optimize network configurations of management by minimizing the sum of location update cost and location paging cost. Experimental results illustrate the effectiveness of the proposed approach. PMID:25180199
A Modified BFGS Formula Using a Trust Region Model for Nonsmooth Convex Minimizations
Cui, Zengru; Yuan, Gonglin; Sheng, Zhou; Liu, Wenjie; Wang, Xiaoliang; Duan, Xiabin
2015-01-01
This paper proposes a modified BFGS formula using a trust region model for solving nonsmooth convex minimizations by using the Moreau-Yosida regularization (smoothing) approach and a new secant equation with a BFGS update formula. Our algorithm uses the function value information and gradient value information to compute the Hessian. The Hessian matrix is updated by the BFGS formula rather than using second-order information of the function, thus decreasing the workload and time involved in the computation. Under suitable conditions, the algorithm converges globally to an optimal solution. Numerical results show that this algorithm can successfully solve nonsmooth unconstrained convex problems. PMID:26501775
On a Highly Nonlinear Self-Obstacle Optimal Control Problem
Di Donato, Daniela; Mugnai, Dimitri
2015-10-15
We consider a non-quadratic optimal control problem associated to a nonlinear elliptic variational inequality, where the obstacle is the control itself. We show that, fixed a desired profile, there exists an optimal solution which is not far from it. Detailed characterizations of the optimal solution are given, also in terms of approximating problems.
Enhanced ant colony optimization for multiscale problems
NASA Astrophysics Data System (ADS)
Hu, Nan; Fish, Jacob
2016-03-01
The present manuscript addresses the issue of computational complexity of optimizing nonlinear composite materials and structures at multiple scales. Several solutions are detailed to meet the enormous computational challenge of optimizing nonlinear structures at multiple scales including: (i) enhanced sampling procedure that provides superior performance of the well-known ant colony optimization algorithm, (ii) a mapping-based meshing of a representative volume element that unlike unstructured meshing permits sensitivity analysis on coarse meshes, and (iii) a multilevel optimization procedure that takes advantage of possible weak coupling of certain scales. We demonstrate the proposed optimization procedure on elastic and inelastic laminated plates involving three scales.
Exact optimal solution for a class of dual control problems
NASA Astrophysics Data System (ADS)
Cao, Suping; Qian, Fucai; Wang, Xiaomei
2016-07-01
This paper considers a discrete-time stochastic optimal control problem for which only measurement equation is partially observed with unknown constant parameters taking value in a finite set of stochastic systems. Because of the fact that the cost-to-go function at each stage contains variance and the non-separability of the variance is so complicated that the dynamic programming cannot be successfully applied, the optimal solution has not been found. In this paper, a new approach to the optimal solution is proposed by embedding the original non-separable problem into a separable auxiliary problem. The theoretical condition on which the optimal solution of the original problem can be attained from a set of solutions of the auxiliary problem is established. In addition, the optimality of the interchanging algorithm is proved and the analytical solution of the optimal control is also obtained. The performance of this controller is illustrated with a simple example.
A weak Hamiltonian finite element method for optimal control problems
NASA Technical Reports Server (NTRS)
Hodges, Dewey H.; Bless, Robert R.
1989-01-01
A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.
A weak Hamiltonian finite element method for optimal control problems
NASA Technical Reports Server (NTRS)
Hodges, Dewey H.; Bless, Robert R.
1990-01-01
A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.
Weak Hamiltonian finite element method for optimal control problems
NASA Technical Reports Server (NTRS)
Hodges, Dewey H.; Bless, Robert R.
1991-01-01
A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.
Wu, Xiaodong
2006-01-01
In this paper, we study several interesting optimal-ratio region detection (ORD) problems in d-D (d ≥ 3) discrete geometric spaces, which arise in high dimensional medical image segmentation. Given a d-D voxel grid of n cells, two classes of geometric regions that are enclosed by a single or two coupled smooth heighfield surfaces defined on the entire grid domain are considered. The objective functions are normalized by a function of the desired regions, which avoids a bias to produce an overly large or small region resulting from data noise. The normalization functions that we employ are used in real medical image segmentation. To our best knowledge, no previous results on these problems in high dimensions are known. We develop a unified algorithmic framework based on a careful characterization of the intrinsic geometric structures and a nontrivial graph transformation scheme, yielding efficient polynomial time algorithms for solving these ORD problems. Our main ideas include the following. We observe that the optimal solution to the ORD problems can be obtained via the construction of a convex hull for a set of O(n) unknown 2-D points using the hand probing technique. The probing oracles are implemented by computing a minimum s-t cut in a weighted directed graph. The ORD problems are then solved by O(n) calls to the minimum s-t cut algorithm. For the class of regions bounded by a single heighfield surface, our further investigation shows that the O(n) calls to the minimum s-t cut algorithm are on a monotone parametric flow network, which enables to detect the optimal-ratio region in the complexity of computing a single maximum flow. PMID:25414538
The Role of Intuition in the Solving of Optimization Problems
ERIC Educational Resources Information Center
Malaspina, Uldarico; Font, Vicenc
2010-01-01
This article presents the partial results obtained in the first stage of the research, which sought to answer the following questions: (a) What is the role of intuition in university students' solutions to optimization problems? (b) What is the role of rigor in university students' solutions to optimization problems? (c) How is the combination of…
Artificial bee colony algorithm for constrained possibilistic portfolio optimization problem
NASA Astrophysics Data System (ADS)
Chen, Wei
2015-07-01
In this paper, we discuss the portfolio optimization problem with real-world constraints under the assumption that the returns of risky assets are fuzzy numbers. A new possibilistic mean-semiabsolute deviation model is proposed, in which transaction costs, cardinality and quantity constraints are considered. Due to such constraints the proposed model becomes a mixed integer nonlinear programming problem and traditional optimization methods fail to find the optimal solution efficiently. Thus, a modified artificial bee colony (MABC) algorithm is developed to solve the corresponding optimization problem. Finally, a numerical example is given to illustrate the effectiveness of the proposed model and the corresponding algorithm.
Execution of Multidisciplinary Design Optimization Approaches on Common Test Problems
NASA Technical Reports Server (NTRS)
Balling, R. J.; Wilkinson, C. A.
1997-01-01
A class of synthetic problems for testing multidisciplinary design optimization (MDO) approaches is presented. These test problems are easy to reproduce because all functions are given as closed-form mathematical expressions. They are constructed in such a way that the optimal value of all variables and the objective is unity. The test problems involve three disciplines and allow the user to specify the number of design variables, state variables, coupling functions, design constraints, controlling design constraints, and the strength of coupling. Several MDO approaches were executed on two sample synthetic test problems. These approaches included single-level optimization approaches, collaborative optimization approaches, and concurrent subspace optimization approaches. Execution results are presented, and the robustness and efficiency of these approaches an evaluated for these sample problems.
Application of the general problem of moments to some optimization problems in elasticity theory
NASA Astrophysics Data System (ADS)
Grigoliuk, E. I.; Fil'Shtinskii, V. A.; Fil'Shtinskii, L. A.
1992-04-01
Several optimization problems in elasticity theory are formulated which are relevant to geomechanics. Methods are then presented for reducing these problems to general moment problems in continuous-function space. By using polynomial approximations of nonstandard moment functions, the general moment problems are reduced to the classical power-law moment problem. This allows an a priori evaluation of the optimal control structure. Theoretical and computational examples are presented.
Computing convex quadrangulations☆
Schiffer, T.; Aurenhammer, F.; Demuth, M.
2012-01-01
We use projected Delaunay tetrahedra and a maximum independent set approach to compute large subsets of convex quadrangulations on a given set of points in the plane. The new method improves over the popular pairing method based on triangulating the point set. PMID:22389540
Optical metrology for very large convex aspheres
NASA Astrophysics Data System (ADS)
Burge, J. H.; Su, P.; Zhao, C.
2008-07-01
Telescopes with very large diameter or with wide fields require convex secondary mirrors that may be many meters in diameter. The optical surfaces for these mirrors can be manufactured to the accuracy limited by the surface metrology. We have developed metrology systems that are specifically optimized for measuring very large convex aspheric surfaces. Large aperture vibration insensitive sub-aperture Fizeau interferometer combined with stitching software give high resolution surface measurements. The global shape is corroborated with a coordinate measuring machine based on the swing arm profilometer.
Multiple local minima in radiotherapy optimization problems with dose-volume constraints.
Deasy, J O
1997-07-01
The cause of multiple local minima in beam weight optimization problems subject to dose-volume constraints is analyzed. Three objective functions were considered: (a) maximization of tumor control probability (TCP), (b) maximization of the minimum target dose, and (c) minimization of the mean-squared-deviation of the target dose from the prescription dose. It is shown that: (a) TCP models generally result in strongly quasiconvex objective functions; (b) maximization of the minimum target dose results in a strongly quasiconvex objective function; and (c) minimizing the root-mean-square dose deviation results in a convex objective function. Dose-volume constraints are considered such that, for each region at risk (RAR), the volume of tissue whose dose exceeds a certain tolerance dose (DTol) is kept equal to or below a given fractional level (VTol). If all RARs lack a "volume effect" (i.e., VTol = 0 for all RARs) then there is a single local minimum. But if volume effects are present, then the feasible space is possibly nonconvex and therefore possibly leads to multiple local minima. These conclusions hold for all three objective functions. Hence, possible local minima come not from the nonlinear nature of the objective functions considered, but from the "either this volume or that volume but not both" nature of the volume effect. These observations imply that optimization algorithms for dose-volume constraint types of problems should have effective strategies for dealing with multiple local minima. PMID:9243478
Solving bi-objective optimal control problems with rectangular framing
NASA Astrophysics Data System (ADS)
Wijaya, Karunia Putra; Götz, Thomas
2016-06-01
Optimization problems, e.g. arising from epidemiology models, often ask for solutions minimizing multi-criteria objective functions. In this paper we discuss a novel approach for solving bi-objective optimal control problems. The set of non-dominated points is constructed via a decreasing sequence of rectangles. Particular attention is paid to a problem with disconnected set of non-dominated points. Several examples from epidemiology are investigated and show the applicability of the method.
Neighboring extremals of dynamic optimization problems with path equality constraints
NASA Technical Reports Server (NTRS)
Lee, A. Y.
1988-01-01
Neighboring extremals of dynamic optimization problems with path equality constraints and with an unknown parameter vector are considered in this paper. With some simplifications, the problem is reduced to solving a linear, time-varying two-point boundary-value problem with integral path equality constraints. A modified backward sweep method is used to solve this problem. Two example problems are solved to illustrate the validity and usefulness of the solution technique.
Optimization neural network for solving flow problems.
Perfetti, R
1995-01-01
This paper describes a neural network for solving flow problems, which are of interest in many areas of application as in fuel, hydro, and electric power scheduling. The neural network consist of two layers: a hidden layer and an output layer. The hidden units correspond to the nodes of the flow graph. The output units represent the branch variables. The network has a linear order of complexity, it is easily programmable, and it is suited for analog very large scale integration (VLSI) realization. The functionality of the proposed network is illustrated by a simulation example concerning the maximal flow problem. PMID:18263420
Singular perturbation analysis of AOTV-related trajectory optimization problems
NASA Technical Reports Server (NTRS)
Calise, Anthony J.; Bae, Gyoung H.
1990-01-01
The problem of real time guidance and optimal control of Aeroassisted Orbit Transfer Vehicles (AOTV's) was addressed using singular perturbation theory as an underlying method of analysis. Trajectories were optimized with the objective of minimum energy expenditure in the atmospheric phase of the maneuver. Two major problem areas were addressed: optimal reentry, and synergetic plane change with aeroglide. For the reentry problem, several reduced order models were analyzed with the objective of optimal changes in heading with minimum energy loss. It was demonstrated that a further model order reduction to a single state model is possible through the application of singular perturbation theory. The optimal solution for the reduced problem defines an optimal altitude profile dependent on the current energy level of the vehicle. A separate boundary layer analysis is used to account for altitude and flight path angle dynamics, and to obtain lift and bank angle control solutions. By considering alternative approximations to solve the boundary layer problem, three guidance laws were derived, each having an analytic feedback form. The guidance laws were evaluated using a Maneuvering Reentry Research Vehicle model and all three laws were found to be near optimal. For the problem of synergetic plane change with aeroglide, a difficult terminal boundary layer control problem arises which to date is found to be analytically intractable. Thus a predictive/corrective solution was developed to satisfy the terminal constraints on altitude and flight path angle. A composite guidance solution was obtained by combining the optimal reentry solution with the predictive/corrective guidance method. Numerical comparisons with the corresponding optimal trajectory solutions show that the resulting performance is very close to optimal. An attempt was made to obtain numerically optimized trajectories for the case where heating rate is constrained. A first order state variable inequality
A Planning Problem Combining Calculus of Variations and Optimal Transport
Carlier, G. Lachapelle, A.
2011-02-15
We consider some variants of the classical optimal transport where not only one optimizes over couplings between some variables x and y but also over some control variables governing the evolutions of these variables with time. Such a situation is motivated by an assignment problem of tasks with workers whose characteristics can evolve with time (and be controlled). We distinguish between the coupled and decoupled case. The coupled case is a standard optimal transport with the value of some optimal control problem as cost. The decoupled case is more involved since it is nonlinear in the transport plan.
Optimality conditions for the numerical solution of optimization problems with PDE constraints :
Aguilo Valentin, Miguel Alejandro; Ridzal, Denis
2014-03-01
A theoretical framework for the numerical solution of partial di erential equation (PDE) constrained optimization problems is presented in this report. This theoretical framework embodies the fundamental infrastructure required to e ciently implement and solve this class of problems. Detail derivations of the optimality conditions required to accurately solve several parameter identi cation and optimal control problems are also provided in this report. This will allow the reader to further understand how the theoretical abstraction presented in this report translates to the application.
Spectral finite-element methods for parametric constrained optimization problems.
Anitescu, M.; Mathematics and Computer Science
2009-01-01
We present a method to approximate the solution mapping of parametric constrained optimization problems. The approximation, which is of the spectral finite element type, is represented as a linear combination of orthogonal polynomials. Its coefficients are determined by solving an appropriate finite-dimensional constrained optimization problem. We show that, under certain conditions, the latter problem is solvable because it is feasible for a sufficiently large degree of the polynomial approximation and has an objective function with bounded level sets. In addition, the solutions of the finite-dimensional problems converge for an increasing degree of the polynomials considered, provided that the solutions exhibit a sufficiently large and uniform degree of smoothness. Our approach solves, in the case of optimization problems with uncertain parameters, the most computationally intensive part of stochastic finite-element approaches. We demonstrate that our framework is applicable to parametric eigenvalue problems.
Statistical properties of convex clustering
Tan, Kean Ming; Witten, Daniela
2016-01-01
In this manuscript, we study the statistical properties of convex clustering. We establish that convex clustering is closely related to single linkage hierarchical clustering and k-means clustering. In addition, we derive the range of the tuning parameter for convex clustering that yields a non-trivial solution. We also provide an unbiased estimator of the degrees of freedom, and provide a finite sample bound for the prediction error for convex clustering. We compare convex clustering to some traditional clustering methods in simulation studies.
Applications of parallel global optimization to mechanics problems
NASA Astrophysics Data System (ADS)
Schutte, Jaco Francois
Global optimization of complex engineering problems, with a high number of variables and local minima, requires sophisticated algorithms with global search capabilities and high computational efficiency. With the growing availability of parallel processing, it makes sense to address these requirements by increasing the parallelism in optimization strategies. This study proposes three methods of concurrent processing. The first method entails exploiting the structure of population-based global algorithms such as the stochastic Particle Swarm Optimization (PSO) algorithm and the Genetic Algorithm (GA). As a demonstration of how such an algorithm may be adapted for concurrent processing we modify and apply the PSO to several mechanical optimization problems on a parallel processing machine. Desirable PSO algorithm features such as insensitivity to design variable scaling and modest sensitivity to algorithm parameters are demonstrated. A second approach to parallelism and improving algorithm efficiency is by utilizing multiple optimizations. With this method a budget of fitness evaluations is distributed among several independent sub-optimizations in place of a single extended optimization. Under certain conditions this strategy obtains a higher combined probability of converging to the global optimum than a single optimization which utilizes the full budget of fitness evaluations. The third and final method of parallelism addressed in this study is the use of quasiseparable decomposition, which is applied to decompose loosely coupled problems. This yields several sub-problems of lesser dimensionality which may be concurrently optimized with reduced effort.
Comparison of optimal design methods in inverse problems
NASA Astrophysics Data System (ADS)
Banks, H. T.; Holm, K.; Kappel, F.
2011-07-01
Typical optimal design methods for inverse or parameter estimation problems are designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates. It is hoped that the inverse problem will produce parameter estimates with increased accuracy using data collected according to the optimal sampling distribution. Here we formulate the classical optimal design problem in the context of general optimization problems over distributions of sampling times. We present a new Prohorov metric-based theoretical framework that permits one to treat succinctly and rigorously any optimal design criteria based on the Fisher information matrix. A fundamental approximation theory is also included in this framework. A new optimal design, SE-optimal design (standard error optimal design), is then introduced in the context of this framework. We compare this new design criterion with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; the standard errors for parameters are computed using asymptotic theory or bootstrapping and the optimal mesh. We use three examples to illustrate ideas: the Verhulst-Pearl logistic population model (Banks H T and Tran H T 2009 Mathematical and Experimental Modeling of Physical and Biological Processes (Boca Raton, FL: Chapman and Hall/CRC)), the standard harmonic oscillator model (Banks H T and Tran H T 2009) and a popular glucose regulation model (Bergman R N, Ider Y Z, Bowden C R and Cobelli C 1979 Am. J. Physiol. 236 E667-77 De Gaetano A and Arino O 2000 J. Math. Biol. 40 136-68 Toffolo G, Bergman R N, Finegood D T, Bowden C R and Cobelli C 1980 Diabetes 29 979-90).
Problems in optimal failure detection and search in radioelectronic equipment
NASA Astrophysics Data System (ADS)
Pashkovskii, G. S.
The book discusses optimal methods for the detection of and search for failures in radioelectronic equipment considered as a system of functionally related elements. Following a review of mathematical models of the problem and methods for the optimization of detection and search procedures, attention is given to optimal methods for the monitoring and diagnostics of systems of high reliability, the monitoring of systems of arbitrary reliability, and the monitoring of systems in the absence of information on system reliability. Methods for the synthesis of optimal automatic monitoring systems are presented, and the choice of solutions under conditions of indeterminancy is examined together with the optimal planning of inspections.
Rapid convergence of airfoil design problems using progressive optimization
NASA Astrophysics Data System (ADS)
Dadone, A.; Grossman, B.
An efficient formulation for the robust design optimization of compressible fluid flow problems is presented. The methodology has three essential ingredients: a highly accurate flow solver, robust and efficient design sensitivities from a discrete adjoint formulation based on a dissipative flow solver and progressive optimization, whereby a sequence of operations, containing a partially converged flow solution, followed by an adjoint solution followed by an optimization step is performed. Furthermore, the progressive optimization involves the use of progressively finer grids. The methodology is shown to be accurate, robust and highly efficient, with a converged design optimization produced in no more than the amount of computational work to perform from one to three flow analyses.
BEM4I applied to shape optimization problems
NASA Astrophysics Data System (ADS)
Zapletal, Jan; Merta, Michal; Čermák, Martin
2016-06-01
Shape optimization problems are one of the areas where the boundary element method can be applied efficiently. We present the application of the BEM4I library developed at IT4Innovations to a class of free surface Bernoulli problems in 3D. Apart from the boundary integral formulation of the related state and adjoint boundary value problems we present an implementation of a general scheme for the treatment of similar problems.
Generalized vector calculus on convex domain
NASA Astrophysics Data System (ADS)
Agrawal, Om P.; Xu, Yufeng
2015-06-01
In this paper, we apply recently proposed generalized integral and differential operators to develop generalized vector calculus and generalized variational calculus for problems defined over a convex domain. In particular, we present some generalization of Green's and Gauss divergence theorems involving some new operators, and apply these theorems to generalized variational calculus. For fractional power kernels, the formulation leads to fractional vector calculus and fractional variational calculus for problems defined over a convex domain. In special cases, when certain parameters take integer values, we obtain formulations for integer order problems. Two examples are presented to demonstrate applications of the generalized variational calculus which utilize the generalized vector calculus developed in the paper. The first example leads to a generalized partial differential equation and the second example leads to a generalized eigenvalue problem, both in two dimensional convex domains. We solve the generalized partial differential equation by using polynomial approximation. A special case of the second example is a generalized isoperimetric problem. We find an approximate solution to this problem. Many physical problems containing integer order integrals and derivatives are defined over arbitrary domains. We speculate that future problems containing fractional and generalized integrals and derivatives in fractional mechanics will be defined over arbitrary domains, and therefore, a general variational calculus incorporating a general vector calculus will be needed for these problems. This research is our first attempt in that direction.
Direct Multiple Shooting Optimization with Variable Problem Parameters
NASA Technical Reports Server (NTRS)
Whitley, Ryan J.; Ocampo, Cesar A.
2009-01-01
Taking advantage of a novel approach to the design of the orbital transfer optimization problem and advanced non-linear programming algorithms, several optimal transfer trajectories are found for problems with and without known analytic solutions. This method treats the fixed known gravitational constants as optimization variables in order to reduce the need for an advanced initial guess. Complex periodic orbits are targeted with very simple guesses and the ability to find optimal transfers in spite of these bad guesses is successfully demonstrated. Impulsive transfers are considered for orbits in both the 2-body frame as well as the circular restricted three-body problem (CRTBP). The results with this new approach demonstrate the potential for increasing robustness for all types of orbit transfer problems.
Group Testing: Four Student Solutions to a Classic Optimization Problem
ERIC Educational Resources Information Center
Teague, Daniel
2006-01-01
This article describes several creative solutions developed by calculus and modeling students to the classic optimization problem of testing in groups to find a small number of individuals who test positive in a large population.
Finding Optimal Gains In Linear-Quadratic Control Problems
NASA Technical Reports Server (NTRS)
Milman, Mark H.; Scheid, Robert E., Jr.
1990-01-01
Analytical method based on Volterra factorization leads to new approximations for optimal control gains in finite-time linear-quadratic control problem of system having infinite number of dimensions. Circumvents need to analyze and solve Riccati equations and provides more transparent connection between dynamics of system and optimal gain.
A Decision Support System for Solving Multiple Criteria Optimization Problems
ERIC Educational Resources Information Center
Filatovas, Ernestas; Kurasova, Olga
2011-01-01
In this paper, multiple criteria optimization has been investigated. A new decision support system (DSS) has been developed for interactive solving of multiple criteria optimization problems (MOPs). The weighted-sum (WS) approach is implemented to solve the MOPs. The MOPs are solved by selecting different weight coefficient values for the criteria…
Enhancements on the Convex Programming Based Powered Descent Guidance Algorithm for Mars Landing
NASA Technical Reports Server (NTRS)
Acikmese, Behcet; Blackmore, Lars; Scharf, Daniel P.; Wolf, Aron
2008-01-01
In this paper, we present enhancements on the powered descent guidance algorithm developed for Mars pinpoint landing. The guidance algorithm solves the powered descent minimum fuel trajectory optimization problem via a direct numerical method. Our main contribution is to formulate the trajectory optimization problem, which has nonconvex control constraints, as a finite dimensional convex optimization problem, specifically as a finite dimensional second order cone programming (SOCP) problem. SOCP is a subclass of convex programming, and there are efficient SOCP solvers with deterministic convergence properties. Hence, the resulting guidance algorithm can potentially be implemented onboard a spacecraft for real-time applications. Particularly, this paper discusses the algorithmic improvements obtained by: (i) Using an efficient approach to choose the optimal time-of-flight; (ii) Using a computationally inexpensive way to detect the feasibility/ infeasibility of the problem due to the thrust-to-weight constraint; (iii) Incorporating the rotation rate of the planet into the problem formulation; (iv) Developing additional constraints on the position and velocity to guarantee no-subsurface flight between the time samples of the temporal discretization; (v) Developing a fuel-limited targeting algorithm; (vi) Initial result on developing an onboard table lookup method to obtain almost fuel optimal solutions in real-time.
Test problem construction for single-objective bilevel optimization.
Sinha, Ankur; Malo, Pekka; Deb, Kalyanmoy
2014-01-01
In this paper, we propose a procedure for designing controlled test problems for single-objective bilevel optimization. The construction procedure is flexible and allows its user to control the different complexities that are to be included in the test problems independently of each other. In addition to properties that control the difficulty in convergence, the procedure also allows the user to introduce difficulties caused by interaction of the two levels. As a companion to the test problem construction framework, the paper presents a standard test suite of 12 problems, which includes eight unconstrained and four constrained problems. Most of the problems are scalable in terms of variables and constraints. To provide baseline results, we have solved the proposed test problems using a nested bilevel evolutionary algorithm. The results can be used for comparison, while evaluating the performance of any other bilevel optimization algorithm. The code related to the paper may be accessed from the website http://bilevel.org . PMID:24364674
Vision-based stereo ranging as an optimal control problem
NASA Technical Reports Server (NTRS)
Menon, P. K. A.; Sridhar, B.; Chatterji, G. B.
1992-01-01
The recent interest in the use of machine vision for flight vehicle guidance is motivated by the need to automate the nap-of-the-earth flight regime of helicopters. Vision-based stereo ranging problem is cast as an optimal control problem in this paper. A quadratic performance index consisting of the integral of the error between observed image irradiances and those predicted by a Pade approximation of the correspondence hypothesis is then used to define an optimization problem. The necessary conditions for optimality yield a set of linear two-point boundary-value problems. These two-point boundary-value problems are solved in feedback form using a version of the backward sweep method. Application of the ranging algorithm is illustrated using a laboratory image pair.
Exact solution for an optimal impermeable parachute problem
NASA Astrophysics Data System (ADS)
Lupu, Mircea; Scheiber, Ernest
2002-10-01
In the paper there are solved direct and inverse boundary problems and analytical solutions are obtained for optimization problems in the case of some nonlinear integral operators. It is modeled the plane potential flow of an inviscid, incompressible and nonlimited fluid jet, witch encounters a symmetrical, curvilinear obstacle--the deflector of maximal drag. There are derived integral singular equations, for direct and inverse problems and the movement in the auxiliary canonical half-plane is obtained. Next, the optimization problem is solved in an analytical manner. The design of the optimal airfoil is performed and finally, numerical computations concerning the drag coefficient and other geometrical and aerodynamical parameters are carried out. This model corresponds to the Helmholtz impermeable parachute problem.
Formulation of the multimission aircraft design optimization problem
NASA Astrophysics Data System (ADS)
Straussfogel, Dennis M.
1998-12-01
The conventional single-mission aircraft design optimization problem is reformulated to allow design and optimization for multiple missions. Defining the aircraft mission as a set of continuous scalar variables leads to the concepts of mission vector and mission space. The multimission aircraft design optimization problem becomes one of optimizing a design for several different points in the mission space, simultaneously. In the limit, a design can be optimized simultaneously for all points in the mission space. Mapping various points from the optimization design space into the mission space generates actual and theoretical optimum performance surfaces. The multimission optimum configuration is defined as the single configuration that minimizes the difference between the actual performance and the theoretical optimum performance. The multimission aircraft design optimization method can be applied over several distinct mission by summing the differences between actual and theoretical optimum performance, or over all possible missions by integrating the difference between the actual and theoretical optimum performance surfaces. The concepts associated with the mission vector, mission space, and multimission optimum configuration are presented in mathematical form. The objective function for the multimission optimization problem is expressed both as a summation over discrete mission points and as an integral over the entire mission space. Mathematical expressions for objective functions based on both single and multiple objectives are developed and presented. A weighting function, emphasizing certain parts of the mission space over others, is also discussed. The multimission aircraft design optimization method is applied to an elementary wing-design optimization problem for an executive jet. The optimization problem is solved numerically for single and multiple objectives, with and without a functional constraint. The effect of the different objective functions and
Climate Intervention as an Optimization Problem
NASA Astrophysics Data System (ADS)
Caldeira, Ken; Ban-Weiss, George A.
2010-05-01
Typically, climate models simulations of intentional intervention in the climate system have taken the approach of imposing a change (eg, in solar flux, aerosol concentrations, aerosol emissions) and then predicting how that imposed change might affect Earth's climate or chemistry. Computations proceed from cause to effect. However, humans often proceed from "What do I want?" to "How do I get it?" One approach to thinking about intentional intervention in the climate system ("geoengineering") is to ask "What kind of climate do we want?" and then ask "What pattern of radiative forcing would come closest to achieving that desired climate state?" This involves defining climate goals and a cost function that measures how closely those goals are attained. (An important next step is to ask "How would we go about producing these desired patterns of radiative forcing?" However, this question is beyond the scope of our present study.) We performed a variety of climate simulations in NCAR's CAM3.1 atmospheric general circulation model with a slab ocean model and thermodynamic sea ice model. We then evaluated, for a specific set of climate forcing basis functions (ie, aerosol concentration distributions), the extent to which the climate response to a linear combination of those basis functions was similar to a linear combination of the climate response to each basis function taken individually. We then developed several cost functions (eg, relative to the 1xCO2 climate, minimize rms difference in zonal and annual mean land temperature, minimize rms difference in zonal and annual mean runoff, minimize rms difference in a combination of these temperature and runoff indices) and then predicted optimal combinations of our basis functions that would minimize these cost functions. Lastly, we produced forward simulations of the predicted optimal radiative forcing patterns and compared these with our expected results. Obviously, our climate model is much simpler than reality and
An object-oriented toolbox for studying optimization problems
NASA Astrophysics Data System (ADS)
Deng, H. Lydia; Gouveia, Wences; Scales, John
The CWP Object-Oriented Optimization Library (COOOL) is a collection of C++ classes for studying and solving optimization problems. It was developed using the freely available GNU compiler gcc. The library contains the basic building blocks for the efficient design of numerical linear algebra and optimization software; it also comes with a variety of unconstrained optimization algorithms and test objective functions drawn from our own research. The only requirement for using one of the optimization methods is that a simple model of communication be followed. This allows us to use exactly the same code to optimize functions tailored for a variety of hardware, no matter what programming language is used. Further, since we have provided class libraries containing building blocks for general purpose optimization and numerical linear algebra software, the development of new algorithms should be greatly aided. COOOL is now freely available via anonymous ftp at
SMMH - A Parallel Heuristic for Combinatorial Optimization Problems
Domingues, Guilherme; Morie, Yoshiyuki; Gu, Feng Long; Nanri, Takeshi; Murakami, Kazuaki
2007-12-26
The process of finding one or more optimal solutions for answering combinatorial optimization problems bases itself on the use of algorithms instances. Those instances usually have to explore a very large search spaces. Heuristics search focusing on the use of High-Order Hopfield neural networks is a largely deployed technique for very large search space. It can be established a very powerful analogy towards the dynamics evolution of a physics spin-glass system while minimizing its own energy and the energy function of the network. This paper presents a new approach for solving combinatorial optimization problems through parallel simulations, based on a High-Order Hopfield neural network using MPI specification.
SMMH--A Parallel Heuristic for Combinatorial Optimization Problems
Domingues, Guilherme; Morie, Yoshiyuki; Gu, Feng Long; Nanri, Takeshi; Murakami, Kazuaki
2007-12-26
The process of finding one or more optimal solutions for answering combinatorial optimization problems bases itself on the use of algorithms instances. Those instances usually have to explore a very large search spaces. Heuristics search focusing on the use of High-Order Hopfield neural networks is a largely deployed technique for very large search space. It can be established a very powerful analogy towards the dynamics evolution of a physics spin-glass system while minimizing its own energy and the energy function of the network. This paper presents a new approach for solving combinatorial optimization problems through parallel simulations, based on a High-Order Hopfield neural network using MPI specification.
Chance-Constrained Guidance With Non-Convex Constraints
NASA Technical Reports Server (NTRS)
Ono, Masahiro
2011-01-01
Missions to small bodies, such as comets or asteroids, require autonomous guidance for descent to these small bodies. Such guidance is made challenging by uncertainty in the position and velocity of the spacecraft, as well as the uncertainty in the gravitational field around the small body. In addition, the requirement to avoid collision with the asteroid represents a non-convex constraint that means finding the optimal guidance trajectory, in general, is intractable. In this innovation, a new approach is proposed for chance-constrained optimal guidance with non-convex constraints. Chance-constrained guidance takes into account uncertainty so that the probability of collision is below a specified threshold. In this approach, a new bounding method has been developed to obtain a set of decomposed chance constraints that is a sufficient condition of the original chance constraint. The decomposition of the chance constraint enables its efficient evaluation, as well as the application of the branch and bound method. Branch and bound enables non-convex problems to be solved efficiently to global optimality. Considering the problem of finite-horizon robust optimal control of dynamic systems under Gaussian-distributed stochastic uncertainty, with state and control constraints, a discrete-time, continuous-state linear dynamics model is assumed. Gaussian-distributed stochastic uncertainty is a more natural model for exogenous disturbances such as wind gusts and turbulence than the previously studied set-bounded models. However, with stochastic uncertainty, it is often impossible to guarantee that state constraints are satisfied, because there is typically a non-zero probability of having a disturbance that is large enough to push the state out of the feasible region. An effective framework to address robustness with stochastic uncertainty is optimization with chance constraints. These require that the probability of violating the state constraints (i.e., the probability of
Josiński, Henryk; Kostrzewa, Daniel; Michalczuk, Agnieszka; Switoński, Adam
2014-01-01
This paper introduces an expanded version of the Invasive Weed Optimization algorithm (exIWO) distinguished by the hybrid strategy of the search space exploration proposed by the authors. The algorithm is evaluated by solving three well-known optimization problems: minimization of numerical functions, feature selection, and the Mona Lisa TSP Challenge as one of the instances of the traveling salesman problem. The achieved results are compared with analogous outcomes produced by other optimization methods reported in the literature. PMID:24955420
Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer
NASA Astrophysics Data System (ADS)
Rosenberg, Gili; Haghnegahdar, Poya; Goddard, Phil; Carr, Peter; Wu, Kesheng; de Prado, Marcos Lopez
2016-09-01
We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.
Solving inverse problems of identification type by optimal control methods
Lenhart, S.; Protopopescu, V.; Yong, J.
1997-05-01
Inverse problems of identification type for nonlinear equations are considered within the framework of optimal control theory. The rigorous solution of any particular problem depends on the functional setting, type of equation, and unknown quantity (or quantities) to be determined. Here we present only the general articulations of the formalism. Compared to classical regularization methods (e.g. Tikhonov coupled with optimization schemes), our approach presents several advantages, namely: (i) a systematic procedure to solve inverse problems of identification type; (ii) an explicit expression for the approximations of the solution; and (iii) a convenient numerical solution of these approximations. {copyright} {ital 1997 American Institute of Physics.}
Solving inverse problems of identification type by optimal control methods
Lenhart, S.; Protopopescu, V.; Jiongmin Yong
1997-06-01
Inverse problems of identification type for nonlinear equations are considered within the framework of optimal control theory. The rigorous solution of any particular problem depends on the functional setting, type of equation, and unknown quantity (or quantities) to be determined. Here the authors present only the general articulations of the formalism. Compared to classical regularization methods (e.g. Tikhonov coupled with optimization schemes), their approach presents several advantages, namely: (i) a systematic procedure to solve inverse problems of identification type; (ii) an explicit expression for the approximations of the solution; and (iii) a convenient numerical solution of these approximations.
Parallel Optimization of Polynomials for Large-scale Problems in Stability and Control
NASA Astrophysics Data System (ADS)
Kamyar, Reza
In this thesis, we focus on some of the NP-hard problems in control theory. Thanks to the converse Lyapunov theory, these problems can often be modeled as optimization over polynomials. To avoid the problem of intractability, we establish a trade off between accuracy and complexity. In particular, we develop a sequence of tractable optimization problems --- in the form of Linear Programs (LPs) and/or Semi-Definite Programs (SDPs) --- whose solutions converge to the exact solution of the NP-hard problem. However, the computational and memory complexity of these LPs and SDPs grow exponentially with the progress of the sequence - meaning that improving the accuracy of the solutions requires solving SDPs with tens of thousands of decision variables and constraints. Setting up and solving such problems is a significant challenge. The existing optimization algorithms and software are only designed to use desktop computers or small cluster computers --- machines which do not have sufficient memory for solving such large SDPs. Moreover, the speed-up of these algorithms does not scale beyond dozens of processors. This in fact is the reason we seek parallel algorithms for setting-up and solving large SDPs on large cluster- and/or super-computers. We propose parallel algorithms for stability analysis of two classes of systems: 1) Linear systems with a large number of uncertain parameters; 2) Nonlinear systems defined by polynomial vector fields. First, we develop a distributed parallel algorithm which applies Polya's and/or Handelman's theorems to some variants of parameter-dependent Lyapunov inequalities with parameters defined over the standard simplex. The result is a sequence of SDPs which possess a block-diagonal structure. We then develop a parallel SDP solver which exploits this structure in order to map the computation, memory and communication to a distributed parallel environment. Numerical tests on a supercomputer demonstrate the ability of the algorithm to
Energy Science and Technology Software Center (ESTSC)
2007-03-01
Aristos is a Trilinos package for nonlinear continuous optimization, based on full-space sequential quadratic programming (SQP) methods. Aristos is specifically designed for the solution of large-scale constrained optimization problems in which the linearized constraint equations require iterative (i.e. inexact) linear solver techniques. Aristos' unique feature is an efficient handling of inexactness in linear system solves. Aristos currently supports the solution of equality-constrained convex and nonconvex optimization problems. It has been used successfully in the areamore » of PDE-constrained optimization, for the solution of nonlinear optimal control, optimal design, and inverse problems.« less
Wang, Chang; Qi, Fei; Shi, Guangming; Wang, Xiaotian
2013-01-01
Deployment is a critical issue affecting the quality of service of camera networks. The deployment aims at adopting the least number of cameras to cover the whole scene, which may have obstacles to occlude the line of sight, with expected observation quality. This is generally formulated as a non-convex optimization problem, which is hard to solve in polynomial time. In this paper, we propose an efficient convex solution for deployment optimizing the observation quality based on a novel anisotropic sensing model of cameras, which provides a reliable measurement of the observation quality. The deployment is formulated as the selection of a subset of nodes from a redundant initial deployment with numerous cameras, which is an ℓ0 minimization problem. Then, we relax this non-convex optimization to a convex ℓ1 minimization employing the sparse representation. Therefore, the high quality deployment is efficiently obtained via convex optimization. Simulation results confirm the effectiveness of the proposed camera deployment algorithms. PMID:23989826
Sub-problem Optimization With Regression and Neural Network Approximators
NASA Technical Reports Server (NTRS)
Guptill, James D.; Hopkins, Dale A.; Patnaik, Surya N.
2003-01-01
Design optimization of large systems can be attempted through a sub-problem strategy. In this strategy, the original problem is divided into a number of smaller problems that are clustered together to obtain a sequence of sub-problems. Solution to the large problem is attempted iteratively through repeated solutions to the modest sub-problems. This strategy is applicable to structures and to multidisciplinary systems. For structures, clustering the substructures generates the sequence of sub-problems. For a multidisciplinary system, individual disciplines, accounting for coupling, can be considered as sub-problems. A sub-problem, if required, can be further broken down to accommodate sub-disciplines. The sub-problem strategy is being implemented into the NASA design optimization test bed, referred to as "CometBoards." Neural network and regression approximators are employed for reanalysis and sensitivity analysis calculations at the sub-problem level. The strategy has been implemented in sequential as well as parallel computational environments. This strategy, which attempts to alleviate algorithmic and reanalysis deficiencies, has the potential to become a powerful design tool. However, several issues have to be addressed before its full potential can be harnessed. This paper illustrates the strategy and addresses some issues.
Lessons Learned During Solutions of Multidisciplinary Design Optimization Problems
NASA Technical Reports Server (NTRS)
Patnaik, Suna N.; Coroneos, Rula M.; Hopkins, Dale A.; Lavelle, Thomas M.
2000-01-01
Optimization research at NASA Glenn Research Center has addressed the design of structures, aircraft and airbreathing propulsion engines. During solution of the multidisciplinary problems several issues were encountered. This paper lists four issues and discusses the strategies adapted for their resolution: (1) The optimization process can lead to an inefficient local solution. This deficiency was encountered during design of an engine component. The limitation was overcome through an augmentation of animation into optimization. (2) Optimum solutions obtained were infeasible for aircraft and air-breathing propulsion engine problems. Alleviation of this deficiency required a cascading of multiple algorithms. (3) Profile optimization of a beam produced an irregular shape. Engineering intuition restored the regular shape for the beam. (4) The solution obtained for a cylindrical shell by a subproblem strategy converged to a design that can be difficult to manufacture. Resolution of this issue remains a challenge. The issues and resolutions are illustrated through six problems: (1) design of an engine component, (2) synthesis of a subsonic aircraft, (3) operation optimization of a supersonic engine, (4) design of a wave-rotor-topping device, (5) profile optimization of a cantilever beam, and (6) design of a cvlindrical shell. The combined effort of designers and researchers can bring the optimization method from academia to industry.
Bicriteria Network Optimization Problem using Priority-based Genetic Algorithm
NASA Astrophysics Data System (ADS)
Gen, Mitsuo; Lin, Lin; Cheng, Runwei
Network optimization is being an increasingly important and fundamental issue in the fields such as engineering, computer science, operations research, transportation, telecommunication, decision support systems, manufacturing, and airline scheduling. In many applications, however, there are several criteria associated with traversing each edge of a network. For example, cost and flow measures are both important in the networks. As a result, there has been recent interest in solving Bicriteria Network Optimization Problem. The Bicriteria Network Optimization Problem is known a NP-hard. The efficient set of paths may be very large, possibly exponential in size. Thus the computational effort required to solve it can increase exponentially with the problem size in the worst case. In this paper, we propose a genetic algorithm (GA) approach used a priority-based chromosome for solving the bicriteria network optimization problem including maximum flow (MXF) model and minimum cost flow (MCF) model. The objective is to find the set of Pareto optimal solutions that give possible maximum flow with minimum cost. This paper also combines Adaptive Weight Approach (AWA) that utilizes some useful information from the current population to readjust weights for obtaining a search pressure toward a positive ideal point. Computer simulations show the several numerical experiments by using some difficult-to-solve network design problems, and show the effectiveness of the proposed method.
Romero, Vicente Jose; Ayon, Douglas V.; Chen, Chun-Hung
2003-09-01
A very general and robust approach to solving optimization problems involving probabilistic uncertainty is through the use of Probabilistic Ordinal Optimization. At each step in the optimization problem, improvement is based only on a relative ranking of the probabilistic merits of local design alternatives, rather than on crisp quantification of the alternatives. Thus, we simply ask the question: 'Is that alternative better or worse than this one?' to some level of statistical confidence we require, not: 'HOW MUCH better or worse is that alternative to this one?'. In this paper we illustrate an elementary application of probabilistic ordinal concepts in a 2-D optimization problem. Two uncertain variables contribute to uncertainty in the response function. We use a simple Coordinate Pattern Search non-gradient-based optimizer to step toward the statistical optimum in the design space. We also discuss more sophisticated implementations, and some of the advantages and disadvantages versus non-ordinal approaches for optimization under uncertainty.
Reliability optimization problems with multiple constraints under fuzziness
NASA Astrophysics Data System (ADS)
Gupta, Neha; Haseen, Sanam; Bari, Abdul
2016-06-01
In reliability optimization problems diverse situation occurs due to which it is not always possible to get relevant precision in system reliability. The imprecision in data can often be represented by triangular fuzzy numbers. In this manuscript, we have considered different fuzzy environment for reliability optimization problem of redundancy. We formulate a redundancy allocation problem for a hypothetical series-parallel system in which the parameters of the system are fuzzy. Two different cases are then formulated as non-linear programming problem and the fuzzy nature is defuzzified into crisp problems using three different defuzzification methods viz. ranking function, graded mean integration value and α-cut. The result of the methods is compared at the end of the manuscript using a numerical example.
Russian Doll Search for solving Constraint Optimization problems
Verfaillie, G.; Lemaitre, M.
1996-12-31
If the Constraint Satisfaction framework has been extended to deal with Constraint Optimization problems, it appears that optimization is far more complex than satisfaction. One of the causes of the inefficiency of complete tree search methods, like Depth First Branch and Bound, lies in the poor quality of the lower bound on the global valuation of a partial assignment, even when using Forward Checking techniques. In this paper, we introduce the Russian Doll Search algorithm which replaces one search by n successive searches on nested subproblems (n being the number of problem variables), records the results of each search and uses them later, when solving larger subproblems, in order to improve the lower bound on the global valuation of any partial assignment. On small random problems and on large real scheduling problems, this algorithm yields surprisingly good results, which greatly improve as the problems get more constrained and the bandwidth of the used variable ordering diminishes.
Salcedo-Sanz, S.; Del Ser, J.; Landa-Torres, I.; Gil-López, S.; Portilla-Figueras, J. A.
2014-01-01
This paper presents a novel bioinspired algorithm to tackle complex optimization problems: the coral reefs optimization (CRO) algorithm. The CRO algorithm artificially simulates a coral reef, where different corals (namely, solutions to the optimization problem considered) grow and reproduce in coral colonies, fighting by choking out other corals for space in the reef. This fight for space, along with the specific characteristics of the corals' reproduction, produces a robust metaheuristic algorithm shown to be powerful for solving hard optimization problems. In this research the CRO algorithm is tested in several continuous and discrete benchmark problems, as well as in practical application scenarios (i.e., optimum mobile network deployment and off-shore wind farm design). The obtained results confirm the excellent performance of the proposed algorithm and open line of research for further application of the algorithm to real-world problems. PMID:25147860
Salcedo-Sanz, S; Del Ser, J; Landa-Torres, I; Gil-López, S; Portilla-Figueras, J A
2014-01-01
This paper presents a novel bioinspired algorithm to tackle complex optimization problems: the coral reefs optimization (CRO) algorithm. The CRO algorithm artificially simulates a coral reef, where different corals (namely, solutions to the optimization problem considered) grow and reproduce in coral colonies, fighting by choking out other corals for space in the reef. This fight for space, along with the specific characteristics of the corals' reproduction, produces a robust metaheuristic algorithm shown to be powerful for solving hard optimization problems. In this research the CRO algorithm is tested in several continuous and discrete benchmark problems, as well as in practical application scenarios (i.e., optimum mobile network deployment and off-shore wind farm design). The obtained results confirm the excellent performance of the proposed algorithm and open line of research for further application of the algorithm to real-world problems. PMID:25147860
Parvizi, A; Van den Broek, W; Koch, C T
2016-04-18
The transport of intensity equation (TIE) is widely applied for recovering wave fronts from an intensity measurement and a measurement of its variation along the direction of propagation. In order to get around the problem of non-uniqueness and ill-conditionedness of the solution of the TIE in the very common case of unspecified boundary conditions or noisy data, additional constraints to the solution are necessary. Although from a numerical optimization point of view, convex constraint as imposed to by total variation minimization is preferable, we will show that in many cases non-convex constraints are necessary to overcome the low-frequency artifacts so typical for convex constraints. We will provide simulated and experimental examples that demonstrate the superiority of solutions to the TIE obtained by our recently introduced gradient flipping algorithm over a total variation constrained solution. PMID:27137272
Block clustering based on difference of convex functions (DC) programming and DC algorithms.
Le, Hoai Minh; Le Thi, Hoai An; Dinh, Tao Pham; Huynh, Van Ngai
2013-10-01
We investigate difference of convex functions (DC) programming and the DC algorithm (DCA) to solve the block clustering problem in the continuous framework, which traditionally requires solving a hard combinatorial optimization problem. DC reformulation techniques and exact penalty in DC programming are developed to build an appropriate equivalent DC program of the block clustering problem. They lead to an elegant and explicit DCA scheme for the resulting DC program. Computational experiments show the robustness and efficiency of the proposed algorithm and its superiority over standard algorithms such as two-mode K-means, two-mode fuzzy clustering, and block classification EM. PMID:23777526
Application of tabu search to deterministic and stochastic optimization problems
NASA Astrophysics Data System (ADS)
Gurtuna, Ozgur
During the past two decades, advances in computer science and operations research have resulted in many new optimization methods for tackling complex decision-making problems. One such method, tabu search, forms the basis of this thesis. Tabu search is a very versatile optimization heuristic that can be used for solving many different types of optimization problems. Another research area, real options, has also gained considerable momentum during the last two decades. Real options analysis is emerging as a robust and powerful method for tackling decision-making problems under uncertainty. Although the theoretical foundations of real options are well-established and significant progress has been made in the theory side, applications are lagging behind. A strong emphasis on practical applications and a multidisciplinary approach form the basic rationale of this thesis. The fundamental concepts and ideas behind tabu search and real options are investigated in order to provide a concise overview of the theory supporting both of these two fields. This theoretical overview feeds into the design and development of algorithms that are used to solve three different problems. The first problem examined is a deterministic one: finding the optimal servicing tours that minimize energy and/or duration of missions for servicing satellites around Earth's orbit. Due to the nature of the space environment, this problem is modeled as a time-dependent, moving-target optimization problem. Two solution methods are developed: an exhaustive method for smaller problem instances, and a method based on tabu search for larger ones. The second and third problems are related to decision-making under uncertainty. In the second problem, tabu search and real options are investigated together within the context of a stochastic optimization problem: option valuation. By merging tabu search and Monte Carlo simulation, a new method for studying options, Tabu Search Monte Carlo (TSMC) method, is
Overcoming the Bellman's curse of dimensionality in large optimization problems
NASA Technical Reports Server (NTRS)
Sobieszczanski-Sobieski, Jaroslaw
1990-01-01
Decomposition of large problems into a hierarchic pyramid of subproblems was proposed in the literature as a means for optimization of engineering systems too large for all-in-one optimization. This decomposition was established heuristically. The dynamic programming (DP) method due to Bellman was augmented with an optimum sensitivity analysis that provides a mathematical basis for the above decomposition, and overcomes the curse of dimensionality that limited the original formulation of DP. Numerical examples are cited.
A general optimality criteria algorithm for a class of engineering optimization problems
NASA Astrophysics Data System (ADS)
Belegundu, Ashok D.
2015-05-01
An optimality criteria (OC)-based algorithm for optimization of a general class of nonlinear programming (NLP) problems is presented. The algorithm is only applicable to problems where the objective and constraint functions satisfy certain monotonicity properties. For multiply constrained problems which satisfy these assumptions, the algorithm is attractive compared with existing NLP methods as well as prevalent OC methods, as the latter involve computationally expensive active set and step-size control strategies. The fixed point algorithm presented here is applicable not only to structural optimization problems but also to certain problems as occur in resource allocation and inventory models. Convergence aspects are discussed. The fixed point update or resizing formula is given physical significance, which brings out a strength and trim feature. The number of function evaluations remains independent of the number of variables, allowing the efficient solution of problems with large number of variables.
Quadratic Optimization in the Problems of Active Control of Sound
NASA Technical Reports Server (NTRS)
Loncaric, J.; Tsynkov, S. V.; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
We analyze the problem of suppressing the unwanted component of a time-harmonic acoustic field (noise) on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate anti-sound. Previously, we have obtained general solutions for active controls in both continuous and discrete formulations of the problem. We have also obtained optimal solutions that minimize the overall absolute acoustic source strength of active control sources. These optimal solutions happen to be particular layers of monopoles on the perimeter of the protected region. Mathematically, minimization of acoustic source strength is equivalent to minimization in the sense of L(sub 1). By contrast. in the current paper we formulate and study optimization problems that involve quadratic functions of merit. Specifically, we minimize the L(sub 2) norm of the control sources, and we consider both the unconstrained and constrained minimization. The unconstrained L(sub 2) minimization is certainly the easiest problem to address numerically. On the other hand, the constrained approach allows one to analyze sophisticated geometries. In a special case, we call compare our finite-difference optimal solutions to the continuous optimal solutions obtained previously using a semi-analytic technique. We also show that the optima obtained in the sense of L(sub 2) differ drastically from those obtained in the sense of L(sub 1).
State-Constrained Optimal Control Problems of Impulsive Differential Equations
Forcadel, Nicolas; Rao Zhiping Zidani, Hasnaa
2013-08-01
The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.
Optimality problem of network topology in stocks market analysis
NASA Astrophysics Data System (ADS)
Djauhari, Maman Abdurachman; Gan, Siew Lee
2015-02-01
Since its introduction fifteen years ago, minimal spanning tree has become an indispensible tool in econophysics. It is to filter the important economic information contained in a complex system of financial markets' commodities. Here we show that, in general, that tool is not optimal in terms of topological properties. Consequently, the economic interpretation of the filtered information might be misleading. To overcome that non-optimality problem, a set of criteria and a selection procedure of an optimal minimal spanning tree will be developed. By using New York Stock Exchange data, the advantages of the proposed method will be illustrated in terms of the power-law of degree distribution.
A Discrete Lagrangian Algorithm for Optimal Routing Problems
Kosmas, O. T.; Vlachos, D. S.; Simos, T. E.
2008-11-06
The ideas of discrete Lagrangian methods for conservative systems are exploited for the construction of algorithms applicable in optimal ship routing problems. The algorithm presented here is based on the discretisation of Hamilton's principle of stationary action Lagrangian and specifically on the direct discretization of the Lagrange-Hamilton principle for a conservative system. Since, in contrast to the differential equations, the discrete Euler-Lagrange equations serve as constrains for the optimization of a given cost functional, in the present work we utilize this feature in order to minimize the cost function for optimal ship routing.
Application of clustering global optimization to thin film design problems.
Lemarchand, Fabien
2014-03-10
Refinement techniques usually calculate an optimized local solution, which is strongly dependent on the initial formula used for the thin film design. In the present study, a clustering global optimization method is used which can iteratively change this initial formula, thereby progressing further than in the case of local optimization techniques. A wide panel of local solutions is found using this procedure, resulting in a large range of optical thicknesses. The efficiency of this technique is illustrated by two thin film design problems, in particular an infrared antireflection coating, and a solar-selective absorber coating. PMID:24663856
The expanded LaGrangian system for constrained optimization problems
NASA Technical Reports Server (NTRS)
Poore, A. B.
1986-01-01
Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrainted optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.
Comparison of Fuzzy Set and Convex Model Theories in Structural Design
NASA Astrophysics Data System (ADS)
Pantelides, Chris P.; Ganzerli, Sara
2001-05-01
A methodology for the treatment of uncertainty in the loads applied to a structural system using convex models is presented and is compared to the fuzzy set finite-element method. The analytical results for a beam, a truss and a frame structure indicate that the two methods based on convex model or fuzzy set theory are in good agreement for equivalent levels of uncertainty applied to linear structures. Convex model or fuzzy set theories have shown that the worst-case scenario response of all possible load combinations cannot be captured simply by load factorisation, as is the current design practice in building codes. Design problems including uncertainty with a large number of degrees of freedom, that are not computationally feasible using conventional methods described in building codes, can be solved easily using convex model or fuzzy set theory. These results can be used directly and efficiently in the analyses required for the optimal design of structural systems, thus enabling optimisation of complex structural systems with uncertainty.
An optimized finite-difference scheme for wave propagation problems
NASA Technical Reports Server (NTRS)
Zingg, D. W.; Lomax, H.; Jurgens, H.
1993-01-01
Two fully-discrete finite-difference schemes for wave propagation problems are presented, a maximum-order scheme and an optimized (or spectral-like) scheme. Both combine a seven-point spatial operator and an explicit six-stage time-march method. The maximum-order operator is fifth-order in space and is sixth-order in time for a linear problem with periodic boundary conditions. The phase and amplitude errors of the schemes obtained using Fourier analysis are given and compared with a second-order and a fourth-order method. Numerical experiments are presented which demonstrate the usefulness of the schemes for a range of problems. For some problems, the optimized scheme leads to a reduction in global error compared to the maximum-order scheme with no additional computational expense.
Artificial Bee Colony Algorithm for Solving Optimal Power Flow Problem
Le Dinh, Luong; Vo Ngoc, Dieu
2013-01-01
This paper proposes an artificial bee colony (ABC) algorithm for solving optimal power flow (OPF) problem. The objective of the OPF problem is to minimize total cost of thermal units while satisfying the unit and system constraints such as generator capacity limits, power balance, line flow limits, bus voltages limits, and transformer tap settings limits. The ABC algorithm is an optimization method inspired from the foraging behavior of honey bees. The proposed algorithm has been tested on the IEEE 30-bus, 57-bus, and 118-bus systems. The numerical results have indicated that the proposed algorithm can find high quality solution for the problem in a fast manner via the result comparisons with other methods in the literature. Therefore, the proposed ABC algorithm can be a favorable method for solving the OPF problem. PMID:24470790
Artificial bee colony algorithm for solving optimal power flow problem.
Le Dinh, Luong; Vo Ngoc, Dieu; Vasant, Pandian
2013-01-01
This paper proposes an artificial bee colony (ABC) algorithm for solving optimal power flow (OPF) problem. The objective of the OPF problem is to minimize total cost of thermal units while satisfying the unit and system constraints such as generator capacity limits, power balance, line flow limits, bus voltages limits, and transformer tap settings limits. The ABC algorithm is an optimization method inspired from the foraging behavior of honey bees. The proposed algorithm has been tested on the IEEE 30-bus, 57-bus, and 118-bus systems. The numerical results have indicated that the proposed algorithm can find high quality solution for the problem in a fast manner via the result comparisons with other methods in the literature. Therefore, the proposed ABC algorithm can be a favorable method for solving the OPF problem. PMID:24470790
NASA Astrophysics Data System (ADS)
Qiu, Zhiping; Wang, Xiaojun
2006-06-01
Two non-probabilistic, set-theoretical methods for determining the maximum and minimum impulsive responses of structures to uncertain-but-bounded impulses are presented. They are, respectively, based on the theories of interval mathematics and convex models. The uncertain-but-bounded impulses are assumed to be a convex set, hyper-rectangle or ellipsoid. For the two non-probabilistic methods, less prior information is required about the uncertain nature of impulses than the probabilistic model. Comparisons between the interval analysis method and the convex model, which are developed as an anti-optimization problem of finding the least favorable impulsive response and the most favorable impulsive response, are made through mathematical analyses and numerical calculations. The results of this study indicate that under the condition of the interval vector being determined from an ellipsoid containing the uncertain impulses, the width of the impulsive responses predicted by the interval analysis method is larger than that by the convex model; under the condition of the ellipsoid being determined from an interval vector containing the uncertain impulses, the width of the interval impulsive responses obtained by the interval analysis method is smaller than that by the convex model.
Identification problem for a wave equation via optimal control
Lenhart, S.; Liang, M.; Protopopescu, V.
1998-11-01
The authors approximate an identification problem by applying optimal control techniques to a Tikhonov`s regularization. They seek to identify the dispersive coefficient in a wave equation and allow for the case of error or uncertainty in the observations used for the identification.
To the optimization problem in minority game model
Yanishevsky, Vasyl
2009-12-14
The article presents the research results of the optimization problem in minority game model to a gaussian approximation using replica symmetry breaking by one step (1RSB). A comparison to replica symmetry approximation (RS) and the results from literary sources received using other methods has been held.
Optimizing Value and Avoiding Problems in Building Schools.
ERIC Educational Resources Information Center
Brevard County School Board, Cocoa, FL.
This report describes school design and construction delivery processes used by the School Board of Brevard County (Cocoa, Florida) that help optimize value, avoid problems, and eliminate the cost of maintaining a large facility staff. The project phases are examined from project definition through design to construction. Project delivery…
Optimal Trees for a Class of Information Retrieval Problems
ERIC Educational Resources Information Center
Stanfel, Larry E.
1973-01-01
A review is given of doubly-chained trees for information retrieval. It is shown that the chaining steps may be of different cost depending upon whether the step is vertical or horizontal. The problem of constructing an optimal retrieval tree is formulated mathematically and a solution procedure given. (5 references) (Author/KE)
Topology optimization for nonlinear dynamic problems: Considerations for automotive crashworthiness
NASA Astrophysics Data System (ADS)
Kaushik, Anshul; Ramani, Anand
2014-04-01
Crashworthiness of automotive structures is most often engineered after an optimal topology has been arrived at using other design considerations. This study is an attempt to incorporate crashworthiness requirements upfront in the topology synthesis process using a mathematically consistent framework. It proposes the use of equivalent linear systems from the nonlinear dynamic simulation in conjunction with a discrete-material topology optimizer. Velocity and acceleration constraints are consistently incorporated in the optimization set-up. Issues specific to crash problems due to the explicit solution methodology employed, nature of the boundary conditions imposed on the structure, etc. are discussed and possible resolutions are proposed. A demonstration of the methodology on two-dimensional problems that address some of the structural requirements and the types of loading typical of frontal and side impact is provided in order to show that this methodology has the potential for topology synthesis incorporating crashworthiness requirements.
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.
Proposal of Evolutionary Simplex Method for Global Optimization Problem
NASA Astrophysics Data System (ADS)
Shimizu, Yoshiaki
To make an agile decision in a rational manner, role of optimization engineering has been notified increasingly under diversified customer demand. With this point of view, in this paper, we have proposed a new evolutionary method serving as an optimization technique in the paradigm of optimization engineering. The developed method has prospects to solve globally various complicated problem appearing in real world applications. It is evolved from the conventional method known as Nelder and Mead’s Simplex method by virtue of idea borrowed from recent meta-heuristic method such as PSO. Mentioning an algorithm to handle linear inequality constraints effectively, we have validated effectiveness of the proposed method through comparison with other methods using several benchmark problems.
Efficient infill sampling for unconstrained robust optimization problems
NASA Astrophysics Data System (ADS)
Rehman, Samee Ur; Langelaar, Matthijs
2016-08-01
A novel infill sampling criterion is proposed for efficient estimation of the global robust optimum of expensive computer simulation based problems. The algorithm is especially geared towards addressing problems that are affected by uncertainties in design variables and problem parameters. The method is based on constructing metamodels using Kriging and adaptively sampling the response surface via a principle of expected improvement adapted for robust optimization. Several numerical examples and an engineering case study are used to demonstrate the ability of the algorithm to estimate the global robust optimum using a limited number of expensive function evaluations.
Optimizing investment fund allocation using vehicle routing problem framework
NASA Astrophysics Data System (ADS)
Mamat, Nur Jumaadzan Zaleha; Jaaman, Saiful Hafizah; Ahmad, Rokiah Rozita
2014-07-01
The objective of investment is to maximize total returns or minimize total risks. To determine the optimum order of investment, vehicle routing problem method is used. The method which is widely used in the field of resource distribution shares almost similar characteristics with the problem of investment fund allocation. In this paper we describe and elucidate the concept of using vehicle routing problem framework in optimizing the allocation of investment fund. To better illustrate these similarities, sectorial data from FTSE Bursa Malaysia is used. Results show that different values of utility for risk-averse investors generate the same investment routes.
Solving Fuzzy Optimization Problem Using Hybrid Ls-Sa Method
NASA Astrophysics Data System (ADS)
Vasant, Pandian
2011-06-01
Fuzzy optimization problem has been one of the most and prominent topics inside the broad area of computational intelligent. It's especially relevant in the filed of fuzzy non-linear programming. It's application as well as practical realization can been seen in all the real world problems. In this paper a large scale non-linear fuzzy programming problem has been solved by hybrid optimization techniques of Line Search (LS), Simulated Annealing (SA) and Pattern Search (PS). As industrial production planning problem with cubic objective function, 8 decision variables and 29 constraints has been solved successfully using LS-SA-PS hybrid optimization techniques. The computational results for the objective function respect to vagueness factor and level of satisfaction has been provided in the form of 2D and 3D plots. The outcome is very promising and strongly suggests that the hybrid LS-SA-PS algorithm is very efficient and productive in solving the large scale non-linear fuzzy programming problem.
Successive linear optimization approach to the dynamic traffic assignment problem
Ho, J.K.
1980-11-01
A dynamic model for the optimal control of traffic flow over a network is considered. The model, which treats congestion explicitly in the flow equations, gives rise to nonlinear, nonconvex mathematical programming problems. It has been shown for a piecewise linear version of this model that a global optimum is contained in the set of optimal solutions of a certain linear program. A sufficient condition for optimality which implies that a global optimum can be obtained by successively optimizing at most N + 1 objective functions for the linear program, where N is the number of time periods in the planning horizon is presented. Computational results are reported to indicate the efficiency of this approach.
An algorithm for computationally expensive engineering optimization problems
NASA Astrophysics Data System (ADS)
Yoel, Tenne
2013-07-01
Modern engineering design often relies on computer simulations to evaluate candidate designs, a scenario which results in an optimization of a computationally expensive black-box function. In these settings, there will often exist candidate designs which cause the simulation to fail, and can therefore degrade the search effectiveness. To address this issue, this paper proposes a new metamodel-assisted computational intelligence optimization algorithm which incorporates classifiers into the optimization search. The classifiers predict which candidate designs are expected to cause the simulation to fail, and this prediction is used to bias the search towards designs predicted to be valid. To enhance the search effectiveness, the proposed algorithm uses an ensemble approach which concurrently employs several metamodels and classifiers. A rigorous performance analysis based on a set of simulation-driven design optimization problems shows the effectiveness of the proposed algorithm.
Statistical physics of hard combinatorial optimization: Vertex cover problem
NASA Astrophysics Data System (ADS)
Zhao, Jin-Hua; Zhou, Hai-Jun
2014-07-01
Typical-case computation complexity is a research topic at the boundary of computer science, applied mathematics, and statistical physics. In the last twenty years, the replica-symmetry-breaking mean field theory of spin glasses and the associated message-passing algorithms have greatly deepened our understanding of typical-case computation complexity. In this paper, we use the vertex cover problem, a basic nondeterministic-polynomial (NP)-complete combinatorial optimization problem of wide application, as an example to introduce the statistical physical methods and algorithms. We do not go into the technical details but emphasize mainly the intuitive physical meanings of the message-passing equations. A nonfamiliar reader shall be able to understand to a large extent the physics behind the mean field approaches and to adjust the mean field methods in solving other optimization problems.
A free boundary approach to shape optimization problems
Bucur, D.; Velichkov, B.
2015-01-01
The analysis of shape optimization problems involving the spectrum of the Laplace operator, such as isoperimetric inequalities, has known in recent years a series of interesting developments essentially as a consequence of the infusion of free boundary techniques. The main focus of this paper is to show how the analysis of a general shape optimization problem of spectral type can be reduced to the analysis of particular free boundary problems. In this survey article, we give an overview of some very recent technical tools, the so-called shape sub- and supersolutions, and show how to use them for the minimization of spectral functionals involving the eigenvalues of the Dirichlet Laplacian, under a volume constraint. PMID:26261362
Optimal Stopping Rules For Some Blackjack Type Problems
NASA Astrophysics Data System (ADS)
Grzybowski, Andrzej Z.
2010-03-01
The paper deals with a class of optimal stopping problems having some features of blackjack type games. A decision maker observes sequentially the values of a finite sequence of non-negative random variables. After each observation he decides whether to stop or to continue. If he decides to stop, he obtains a payoff dependent on the sum of already observed values. The greater the sum, the more the decision maker gains, unless the sum exceeds a positive number T-a limit given in the problem. If so, the decision maker loses all or part of his payoff. A sufficient condition for existence of a simple optimal stopping rule for such problems is formulated. Then some special cases are considered in detail. Some numerical examples and practical questions are discussed as well.
Duan, Hai-Bin; Xu, Chun-Fang; Xing, Zhi-Hui
2010-02-01
In this paper, a novel hybrid Artificial Bee Colony (ABC) and Quantum Evolutionary Algorithm (QEA) is proposed for solving continuous optimization problems. ABC is adopted to increase the local search capacity as well as the randomness of the populations. In this way, the improved QEA can jump out of the premature convergence and find the optimal value. To show the performance of our proposed hybrid QEA with ABC, a number of experiments are carried out on a set of well-known Benchmark continuous optimization problems and the related results are compared with two other QEAs: the QEA with classical crossover operation, and the QEA with 2-crossover strategy. The experimental comparison results demonstrate that the proposed hybrid ABC and QEA approach is feasible and effective in solving complex continuous optimization problems. PMID:20180252
Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures
NASA Astrophysics Data System (ADS)
Flaßkamp, Kathrin; Ober-Blöbaum, Sina; Kobilarov, Marin
2012-08-01
Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.
Block-oriented modeling of superstructure optimization problems
Friedman, Z; Ingalls, J; Siirola, JD; Watson, JP
2013-10-15
We present a novel software framework for modeling large-scale engineered systems as mathematical optimization problems. A key motivating feature in such systems is their hierarchical, highly structured topology. Existing mathematical optimization modeling environments do not facilitate the natural expression and manipulation of hierarchically structured systems. Rather, the modeler is forced to "flatten" the system description, hiding structure that may be exploited by solvers, and obfuscating the system that the modeling environment is attempting to represent. To correct this deficiency, we propose a Python-based "block-oriented" modeling approach for representing the discrete components within the system. Our approach is an extension of the Pyomo library for specifying mathematical optimization problems. Through the use of a modeling components library, the block-oriented approach facilitates a clean separation of system superstructure from the details of individual components. This approach also naturally lends itself to expressing design and operational decisions as disjunctive expressions over the component blocks. By expressing a mathematical optimization problem in a block-oriented manner, inherent structure (e.g., multiple scenarios) is preserved for potential exploitation by solvers. In particular, we show that block-structured mathematical optimization problems can be straightforwardly manipulated by decomposition-based multi-scenario algorithmic strategies, specifically in the context of the PySP stochastic programming library. We illustrate our block-oriented modeling approach using a case study drawn from the electricity grid operations domain: unit commitment with transmission switching and N - 1 reliability constraints. Finally, we demonstrate that the overhead associated with block-oriented modeling only minimally increases model instantiation times, and need not adversely impact solver behavior. (C) 2013 Elsevier Ltd. All rights reserved.
A convex complementarity approach for simulating large granular flows.
Tasora, A.; Anitescu, M.; Mathematics and Computer Science; Univ. degli Studi di Parma
2010-07-01
Aiming at the simulation of dense granular flows, we propose and test a numerical method based on successive convex complementarity problems. This approach originates from a multibody description of the granular flow: all the particles are simulated as rigid bodies with arbitrary shapes and frictional contacts. Unlike the discrete element method (DEM), the proposed approach does not require small integration time steps typical of stiff particle interaction; this fact, together with the development of optimized algorithms that can run also on parallel computing architectures, allows an efficient application of the proposed methodology to granular flows with a large number of particles. We present an application to the analysis of the refueling flow in pebble-bed nuclear reactors. Extensive validation of our method against both DEM and physical experiments results indicates that essential collective characteristics of dense granular flow are accurately predicted.
Multiresolution strategies for the numerical solution of optimal control problems
NASA Astrophysics Data System (ADS)
Jain, Sachin
There exist many numerical techniques for solving optimal control problems but less work has been done in the field of making these algorithms run faster and more robustly. The main motivation of this work is to solve optimal control problems accurately in a fast and efficient way. Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. The algorithm adapted dynamically to any existing or emerging irregularities in the solution by automatically allocating more grid points to the region where the solution exhibited sharp features and fewer points to the region where the solution was smooth. Thereby, the computational time and memory usage has been reduced significantly, while maintaining an accuracy equivalent to the one obtained using a fine uniform mesh. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a
Analog and digital FPGA implementation of BRIN for optimization problems.
Ng, H S; Lam, K P
2003-01-01
The binary relation inference network (BRIN) shows promise in obtaining the global optimal solution for optimization problem, which is time independent of the problem size. However, the realization of this method is dependent on the implementation platforms. We studied analog and digital FPGA implementation platforms. Analog implementation of BRIN for two different directed graph problems is studied. As transitive closure problems can transform to a special case of shortest path problems or a special case of maximum spanning tree problems, two different forms of BRIN are discussed. Their circuits using common analog integrated circuits are investigated. The BRIN solution for critical path problems is expressed and is implemented using the separated building block circuit and the combined building block circuit. As these circuits are different, the response time of these networks will be different. The advancement of field programmable gate arrays (FPGAs) in recent years, allowing millions of gates on a single chip and accompanying with high-level design tools, has allowed the implementation of very complex networks. With this exemption on manual circuit construction and availability of efficient design platform, the BRIN architecture could be built in a much more efficient way. Problems on bandwidth are removed by taking all previous external connections to the inside of the chip. By transforming BRIN to FPGA (Xilinx XC4010XL and XCV800 Virtex), we implement a synchronous network with computations in a finite number of steps. Two case studies are presented, with correct results verified from simulation implementation. Resource consumption on FPGAs is studied showing that Virtex devices are more suitable for the expansion of network in future developments. PMID:18244587
Rigorous location of phase transitions in hard optimization problems.
Achlioptas, Dimitris; Naor, Assaf; Peres, Yuval
2005-06-01
It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme 'hardness', computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm. PMID:15944693
Tunneling and Speedup in Permutation-Invariant Quantum Optimization Problem
NASA Astrophysics Data System (ADS)
Albash, Tameem
Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via the quantum adiabatic algorithm. Restricting ourselves to qubit-permutation invariant problems, we show that tunneling in these problems can be understood using the semi-classical potential derived from the spin-coherent path integral formalism. Using this, we show that the class of problems that fall under Reichardt's bound (1), i.e., have a constant gap and hence can be efficiently solved using the quantum adiabatic algorithm, do not exhibit tunneling in the large system-size limit. We proceed to construct problems that do not fall under Reichardt's bound but numerically have a constant gap and do exhibit tunneling. However, perhaps counter-intuitively, tunneling does not provide the most efficient mechanism for finding the solution to these problems. Instead, an evolution involving a sequence of diabatic transitions through many avoided level-crossings, involving no tunneling, is optimal and outperforms tunneling in the adiabatic regime. In yet another twist, we show that in this case, classical spin-vector dynamics is as efficient as the diabatic quantum evolution (2).
A matrix product state method for solving combinatorial optimization problems
NASA Astrophysics Data System (ADS)
Pelton, S. S.; Chamon, C.; Mucciolo, E. R.
2015-03-01
We present a method based on a matrix product state representation to solve combinatorial optimization problems. All constraints are met by mapping Boolean gates into projection operators and applying operators sequentially. The method provides exact solutions with high success probability, even in the case of frustrated systems. The computational cost of the method is controlled by the maximum relative entropy of the system. Results of numerical simulations for several types of problems will be shown and discussed. NSF Grants CCF-1116590 and CCF-1117241.
Performance of quantum annealing in solving optimization problems: A review
NASA Astrophysics Data System (ADS)
Suzuki, S.
2015-02-01
Quantum annealing is one of the optimization method for generic optimization problems. It uses quantum mechanics and is implemented by a quantum computer ideally. At the earlier stage, several numerical experiments using conventional computers have provided results showing that quantum annealing produces an answer faster than simulated annealing, a classical counterpart of quantum annealing. Later, theoretical and numerical studies have shown that there are drawbacks in quantum annealing. The power of quantum annealing is still an open problem. What makes quantum annealing a hot topic now is that a quantum computer based on quantum annealing is manufactured and commercialized by a Canadian company named D-Wave Systems. In the present article, we review the study of quantum annealing, focusing mainly on its power.
Fuel-optimal trajectories for aeroassisted coplanar orbital transfer problem
NASA Technical Reports Server (NTRS)
Naidu, Desineni Subbaramaiah; Hibey, Joseph L.; Charalambous, Charalambos D.
1990-01-01
The optimal control problem arising in coplanar orbital transfer employing aeroassist technology is addressed. The maneuver involves the transfer from high to low earth orbit via the atmosphere, with the object of minimizing the total fuel consumption. Simulations are carried out to obtain the fuel-optimal trajectories for flying the spacecraft through the atmosphere. A highlight is the application of an efficient multiple-shooting method for treating the nonlinear two-point boundary value problem resulting from the optimizaion procedure. The strategy for the atmospheric portion of the minimum-fuel transfer is to fly at the maximum lift-to-drag ratio L/D initially in order to recover from the downward plunge, and then to fly at a negative L/D to level off the flight so that the vehicle skips out of the atmosphere with a flight path angle near zero degrees.
Fuel-optimal trajectories for aeroassisted coplanar orbital transfer problem
NASA Astrophysics Data System (ADS)
Naidu, Desineni Subbaramaiah; Hibey, Joseph L.; Charalambous, Charalambos D.
1990-03-01
The optimal control problem arising in coplanar orbital transfer employing aeroassist technology is addressed. The maneuver involves the transfer from high to low earth orbit via the atmosphere, with the object of minimizing the total fuel consumption. Simulations are carried out to obtain the fuel-optimal trajectories for flying the spacecraft through the atmosphere. A highlight is the application of an efficient multiple-shooting method for treating the nonlinear two-point boundary value problem resulting from the optimizaion procedure. The strategy for the atmospheric portion of the minimum-fuel transfer is to fly at the maximum lift-to-drag ratio L/D initially in order to recover from the downward plunge, and then to fly at a negative L/D to level off the flight so that the vehicle skips out of the atmosphere with a flight path angle near zero degrees.
An optimized spectral difference scheme for CAA problems
NASA Astrophysics Data System (ADS)
Gao, Junhui; Yang, Zhigang; Li, Xiaodong
2012-05-01
In the implementation of spectral difference (SD) method, the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes. The errors incurred in using extrapolation and interpolation would result in instability. On the other hand, the difference between the left and right conserved variables at the edge interface will introduce dissipation to the SD method when applying a Riemann solver to compute the flux at the element interface. In this paper, an optimization of the extrapolation and interpolation schemes for the fourth order SD method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers. The optimized coefficients of the extrapolation and interpolation are presented. And the dispersion error of the original and optimized schemes is plotted and compared. An improvement of the dispersion error over the resolvable wavenumber range of SD method is obtained. The stability of the optimized fourth order SD scheme is analyzed. It is found that the stability of the 4th order scheme with Chebyshev-Gauss-Lobatto flux points, which is originally weakly unstable, has been improved through the optimization. The weak instability is eliminated completely if an additional second order filter is applied on selected flux points. One and two dimensional linear wave propagation analyses are carried out for the optimized scheme. It is found that in the resolvable wavenumber range the new SD scheme is less dispersive and less dissipative than the original scheme, and the new scheme is less anisotropic for 2D wave propagation. The optimized SD solver is validated with four computational aeroacoustics (CAA) workshop benchmark problems. The numerical results with optimized schemes agree much better with the analytical data than those with the original schemes.
Asynchronous global optimization techniques for medium and large inversion problems
Pereyra, V.; Koshy, M.; Meza, J.C.
1995-04-01
We discuss global optimization procedures adequate for seismic inversion problems. We explain how to save function evaluations (which may involve large scale ray tracing or other expensive operations) by creating a data base of information on what parts of parameter space have already been inspected. It is also shown how a correct parallel implementation using PVM speeds up the process almost linearly with respect to the number of processors, provided that the function evaluations are expensive enough to offset the communication overhead.
Convex polytopes and quantum separability
Holik, F.; Plastino, A.
2011-12-15
We advance a perspective of the entanglement issue that appeals to the Schlienz-Mahler measure [Phys. Rev. A 52, 4396 (1995)]. Related to it, we propose a criterium based on the consideration of convex subsets of quantum states. This criterium generalizes a property of product states to convex subsets (of the set of quantum states) that is able to uncover an interesting geometrical property of the separability property.
Convex polytopes and quantum separability
NASA Astrophysics Data System (ADS)
Holik, F.; Plastino, A.
2011-12-01
We advance a perspective of the entanglement issue that appeals to the Schlienz-Mahler measure [Phys. Rev. APLRAAN1050-294710.1103/PhysRevA.52.4396 52, 4396 (1995)]. Related to it, we propose a criterium based on the consideration of convex subsets of quantum states. This criterium generalizes a property of product states to convex subsets (of the set of quantum states) that is able to uncover an interesting geometrical property of the separability property.
Two Error Bounds for Constrained Optimization Problems and Their Applications
Wang Changyu Zhang Jianzhong; Zhao Wenling
2008-06-15
This paper presents a global error bound for the projected gradient and a local error bound for the distance from a feasible solution to the optimal solution set of a nonlinear programming problem by using some characteristic quantities such as value function, trust region radius etc., which are appeared in the trust region method. As applications of these error bounds, we obtain sufficient conditions under which a sequence of feasible solutions converges to a stationary point or to an optimal solution, respectively, and a necessary and sufficient condition under which a sequence of feasible solutions converges to a Kuhn-Tucker point. Other applications involve finite termination of a sequence of feasible solutions. For general optimization problems, when the optimal solution set is generalized non-degenerate or gives generalized weak sharp minima, we give a necessary and sufficient condition for a sequence of feasible solutions to terminate finitely at a Kuhn-Tucker point, and a sufficient condition which guarantees that a sequence of feasible solutions terminates finitely at a stationary point.
Solving nonlinear equality constrained multiobjective optimization problems using neural networks.
Mestari, Mohammed; Benzirar, Mohammed; Saber, Nadia; Khouil, Meryem
2015-10-01
This paper develops a neural network architecture and a new processing method for solving in real time, the nonlinear equality constrained multiobjective optimization problem (NECMOP), where several nonlinear objective functions must be optimized in a conflicting situation. In this processing method, the NECMOP is converted to an equivalent scalar optimization problem (SOP). The SOP is then decomposed into several-separable subproblems processable in parallel and in a reasonable time by multiplexing switched capacitor circuits. The approach which we propose makes use of a decomposition-coordination principle that allows nonlinearity to be treated at a local level and where coordination is achieved through the use of Lagrange multipliers. The modularity and the regularity of the neural networks architecture herein proposed make it suitable for very large scale integration implementation. An application to the resolution of a physical problem is given to show that the approach used here possesses some advantages of the point of algorithmic view, and provides processes of resolution often simpler than the usual techniques. PMID:25647664
Issues and Strategies in Solving Multidisciplinary Optimization Problems
NASA Technical Reports Server (NTRS)
Patnaik, Surya
2013-01-01
Optimization research at NASA Glenn Research Center has addressed the design of structures, aircraft and airbreathing propulsion engines. The accumulated multidisciplinary design activity is collected under a testbed entitled COMETBOARDS. Several issues were encountered during the solution of the problems. Four issues and the strategies adapted for their resolution are discussed. This is followed by a discussion on analytical methods that is limited to structural design application. An optimization process can lead to an inefficient local solution. This deficiency was encountered during design of an engine component. The limitation was overcome through an augmentation of animation into optimization. Optimum solutions obtained were infeasible for aircraft and airbreathing propulsion engine problems. Alleviation of this deficiency required a cascading of multiple algorithms. Profile optimization of a beam produced an irregular shape. Engineering intuition restored the regular shape for the beam. The solution obtained for a cylindrical shell by a subproblem strategy converged to a design that can be difficult to manufacture. Resolution of this issue remains a challenge. The issues and resolutions are illustrated through a set of problems: Design of an engine component, Synthesis of a subsonic aircraft, Operation optimization of a supersonic engine, Design of a wave-rotor-topping device, Profile optimization of a cantilever beam, and Design of a cylindrical shell. This chapter provides a cursory account of the issues. Cited references provide detailed discussion on the topics. Design of a structure can also be generated by traditional method and the stochastic design concept. Merits and limitations of the three methods (traditional method, optimization method and stochastic concept) are illustrated. In the traditional method, the constraints are manipulated to obtain the design and weight is back calculated. In design optimization, the weight of a structure becomes the
Wu, Zong-Sheng; Fu, Wei-Ping; Xue, Ru
2015-01-01
Teaching-learning-based optimization (TLBO) algorithm is proposed in recent years that simulates the teaching-learning phenomenon of a classroom to effectively solve global optimization of multidimensional, linear, and nonlinear problems over continuous spaces. In this paper, an improved teaching-learning-based optimization algorithm is presented, which is called nonlinear inertia weighted teaching-learning-based optimization (NIWTLBO) algorithm. This algorithm introduces a nonlinear inertia weighted factor into the basic TLBO to control the memory rate of learners and uses a dynamic inertia weighted factor to replace the original random number in teacher phase and learner phase. The proposed algorithm is tested on a number of benchmark functions, and its performance comparisons are provided against the basic TLBO and some other well-known optimization algorithms. The experiment results show that the proposed algorithm has a faster convergence rate and better performance than the basic TLBO and some other algorithms as well. PMID:26421005
Optimal Parametric Discrete Event Control: Problem and Solution
Griffin, Christopher H
2008-01-01
We present a novel optimization problem for discrete event control, similar in spirit to the optimal parametric control problem common in statistical process control. In our problem, we assume a known finite state machine plant model $G$ defined over an event alphabet $\\Sigma$ so that the plant model language $L = \\LanM(G)$ is prefix closed. We further assume the existence of a \\textit{base control structure} $M_K$, which may be either a finite state machine or a deterministic pushdown machine. If $K = \\LanM(M_K)$, we assume $K$ is prefix closed and that $K \\subseteq L$. We associate each controllable transition of $M_K$ with a binary variable $X_1,\\dots,X_n$ indicating whether the transition is enabled or not. This leads to a function $M_K(X_1,\\dots,X_n)$, that returns a new control specification depending upon the values of $X_1,\\dots,X_n$. We exhibit a branch-and-bound algorithm to solve the optimization problem $\\min_{X_1,\\dots,X_n}\\max_{w \\in K} C(w)$ such that $M_K(X_1,\\dots,X_n) \\models \\Pi$ and $\\LanM(M_K(X_1,\\dots,X_n)) \\in \\Con(L)$. Here $\\Pi$ is a set of logical assertions on the structure of $M_K(X_1,\\dots,X_n)$, and $M_K(X_1,\\dots,X_n) \\models \\Pi$ indicates that $M_K(X_1,\\dots,X_n)$ satisfies the logical assertions; and, $\\Con(L)$ is the set of controllable sublanguages of $L$.
A Memetic Algorithm for Global Optimization of Multimodal Nonseparable Problems.
Zhang, Geng; Li, Yangmin
2016-06-01
It is a big challenging issue of avoiding falling into local optimum especially when facing high-dimensional nonseparable problems where the interdependencies among vector elements are unknown. In order to improve the performance of optimization algorithm, a novel memetic algorithm (MA) called cooperative particle swarm optimizer-modified harmony search (CPSO-MHS) is proposed in this paper, where the CPSO is used for local search and the MHS for global search. The CPSO, as a local search method, uses 1-D swarm to search each dimension separately and thus converges fast. Besides, it can obtain global optimum elements according to our experimental results and analyses. MHS implements the global search by recombining different vector elements and extracting global optimum elements. The interaction between local search and global search creates a set of local search zones, where global optimum elements reside within the search space. The CPSO-MHS algorithm is tested and compared with seven other optimization algorithms on a set of 28 standard benchmarks. Meanwhile, some MAs are also compared according to the results derived directly from their corresponding references. The experimental results demonstrate a good performance of the proposed CPSO-MHS algorithm in solving multimodal nonseparable problems. PMID:26292352
An Efficient Optimization Method for Solving Unsupervised Data Classification Problems
Shabanzadeh, Parvaneh; Yusof, Rubiyah
2015-01-01
Unsupervised data classification (or clustering) analysis is one of the most useful tools and a descriptive task in data mining that seeks to classify homogeneous groups of objects based on similarity and is used in many medical disciplines and various applications. In general, there is no single algorithm that is suitable for all types of data, conditions, and applications. Each algorithm has its own advantages, limitations, and deficiencies. Hence, research for novel and effective approaches for unsupervised data classification is still active. In this paper a heuristic algorithm, Biogeography-Based Optimization (BBO) algorithm, was adapted for data clustering problems by modifying the main operators of BBO algorithm, which is inspired from the natural biogeography distribution of different species. Similar to other population-based algorithms, BBO algorithm starts with an initial population of candidate solutions to an optimization problem and an objective function that is calculated for them. To evaluate the performance of the proposed algorithm assessment was carried on six medical and real life datasets and was compared with eight well known and recent unsupervised data classification algorithms. Numerical results demonstrate that the proposed evolutionary optimization algorithm is efficient for unsupervised data classification. PMID:26336509
An Efficient Optimization Method for Solving Unsupervised Data Classification Problems.
Shabanzadeh, Parvaneh; Yusof, Rubiyah
2015-01-01
Unsupervised data classification (or clustering) analysis is one of the most useful tools and a descriptive task in data mining that seeks to classify homogeneous groups of objects based on similarity and is used in many medical disciplines and various applications. In general, there is no single algorithm that is suitable for all types of data, conditions, and applications. Each algorithm has its own advantages, limitations, and deficiencies. Hence, research for novel and effective approaches for unsupervised data classification is still active. In this paper a heuristic algorithm, Biogeography-Based Optimization (BBO) algorithm, was adapted for data clustering problems by modifying the main operators of BBO algorithm, which is inspired from the natural biogeography distribution of different species. Similar to other population-based algorithms, BBO algorithm starts with an initial population of candidate solutions to an optimization problem and an objective function that is calculated for them. To evaluate the performance of the proposed algorithm assessment was carried on six medical and real life datasets and was compared with eight well known and recent unsupervised data classification algorithms. Numerical results demonstrate that the proposed evolutionary optimization algorithm is efficient for unsupervised data classification. PMID:26336509
Adjoint optimization of natural convection problems: differentially heated cavity
NASA Astrophysics Data System (ADS)
Saglietti, Clio; Schlatter, Philipp; Monokrousos, Antonios; Henningson, Dan S.
2016-06-01
Optimization of natural convection-driven flows may provide significant improvements to the performance of cooling devices, but a theoretical investigation of such flows has been rarely done. The present paper illustrates an efficient gradient-based optimization method for analyzing such systems. We consider numerically the natural convection-driven flow in a differentially heated cavity with three Prandtl numbers (Pr=0.15{-}7 ) at super-critical conditions. All results and implementations were done with the spectral element code Nek5000. The flow is analyzed using linear direct and adjoint computations about a nonlinear base flow, extracting in particular optimal initial conditions using power iteration and the solution of the full adjoint direct eigenproblem. The cost function for both temperature and velocity is based on the kinetic energy and the concept of entransy, which yields a quadratic functional. Results are presented as a function of Prandtl number, time horizons and weights between kinetic energy and entransy. In particular, it is shown that the maximum transient growth is achieved at time horizons on the order of 5 time units for all cases, whereas for larger time horizons the adjoint mode is recovered as optimal initial condition. For smaller time horizons, the influence of the weights leads either to a concentric temperature distribution or to an initial condition pattern that opposes the mean shear and grows according to the Orr mechanism. For specific cases, it could also been shown that the computation of optimal initial conditions leads to a degenerate problem, with a potential loss of symmetry. In these situations, it turns out that any initial condition lying in a specific span of the eigenfunctions will yield exactly the same transient amplification. As a consequence, the power iteration converges very slowly and fails to extract all possible optimal initial conditions. According to the authors' knowledge, this behavior is illustrated here
Mathematical theory of a relaxed design problem in structural optimization
NASA Technical Reports Server (NTRS)
Kikuchi, Noboru; Suzuki, Katsuyuki
1990-01-01
Various attempts have been made to construct a rigorous mathematical theory of optimization for size, shape, and topology (i.e. layout) of an elastic structure. If these are represented by a finite number of parametric functions, as Armand described, it is possible to construct an existence theory of the optimum design using compactness argument in a finite dimensional design space or a closed admissible set of a finite dimensional design space. However, if the admissible design set is a subset of non-reflexive Banach space such as L(sup infinity)(Omega), construction of the existence theory of the optimum design becomes suddenly difficult and requires to extend (i.e. generalize) the design problem to much more wider class of design that is compatible to mechanics of structures in the sense of variational principle. Starting from the study by Cheng and Olhoff, Lurie, Cherkaev, and Fedorov introduced a new concept of convergence of design variables in a generalized sense and construct the 'G-Closure' theory of an extended (relaxed) optimum design problem. A similar attempt, but independent in large extent, can also be found in Kohn and Strang in which the shape and topology optimization problem is relaxed to allow to use of perforated composites rather than restricting it to usual solid structures. An identical idea is also stated in Murat and Tartar using the notion of the homogenization theory. That is, introducing possibility of micro-scale perforation together with the theory of homogenization, the optimum design problem is relaxed to construct its mathematical theory. It is also noted that this type of relaxed design problem is perfectly matched to the variational principle in structural mechanics.
On the robust optimization to the uncertain vaccination strategy problem
Chaerani, D. Anggriani, N. Firdaniza
2014-02-21
In order to prevent an epidemic of infectious diseases, the vaccination coverage needs to be minimized and also the basic reproduction number needs to be maintained below 1. This means that as we get the vaccination coverage as minimum as possible, thus we need to prevent the epidemic to a small number of people who already get infected. In this paper, we discuss the case of vaccination strategy in term of minimizing vaccination coverage, when the basic reproduction number is assumed as an uncertain parameter that lies between 0 and 1. We refer to the linear optimization model for vaccination strategy that propose by Becker and Starrzak (see [2]). Assuming that there is parameter uncertainty involved, we can see Tanner et al (see [9]) who propose the optimal solution of the problem using stochastic programming. In this paper we discuss an alternative way of optimizing the uncertain vaccination strategy using Robust Optimization (see [3]). In this approach we assume that the parameter uncertainty lies within an ellipsoidal uncertainty set such that we can claim that the obtained result will be achieved in a polynomial time algorithm (as it is guaranteed by the RO methodology). The robust counterpart model is presented.
On the robust optimization to the uncertain vaccination strategy problem
NASA Astrophysics Data System (ADS)
Chaerani, D.; Anggriani, N.; Firdaniza
2014-02-01
In order to prevent an epidemic of infectious diseases, the vaccination coverage needs to be minimized and also the basic reproduction number needs to be maintained below 1. This means that as we get the vaccination coverage as minimum as possible, thus we need to prevent the epidemic to a small number of people who already get infected. In this paper, we discuss the case of vaccination strategy in term of minimizing vaccination coverage, when the basic reproduction number is assumed as an uncertain parameter that lies between 0 and 1. We refer to the linear optimization model for vaccination strategy that propose by Becker and Starrzak (see [2]). Assuming that there is parameter uncertainty involved, we can see Tanner et al (see [9]) who propose the optimal solution of the problem using stochastic programming. In this paper we discuss an alternative way of optimizing the uncertain vaccination strategy using Robust Optimization (see [3]). In this approach we assume that the parameter uncertainty lies within an ellipsoidal uncertainty set such that we can claim that the obtained result will be achieved in a polynomial time algorithm (as it is guaranteed by the RO methodology). The robust counterpart model is presented.
Optimality conditions for a two-stage reservoir operation problem
NASA Astrophysics Data System (ADS)
Zhao, Jianshi; Cai, Ximing; Wang, Zhongjing
2011-08-01
This paper discusses the optimality conditions for standard operation policy (SOP) and hedging rule (HR) for a two-stage reservoir operation problem using a consistent theoretical framework. The effects of three typical constraints, i.e., mass balance, nonnegative release, and storage constraints under both certain and uncertain conditions are analyzed. When all nonnegative constraints and storage constraints are unbinding, HR results in optimal reservoir operation following the marginal benefit (MB) principle (the MB is equal over current and future stages. However, if any of those constraints are binding, SOP results in the optimal solution, except in some special cases which need to carry over water in the current stage to the future stage, when extreme drought is certain and a higher marginal utility exists for the second stage. Furthermore, uncertainty complicates the effects of the various constraints. A higher uncertainty level in the future makes HR more favorable as water needs to be reserved to defend against the risk caused by uncertainty. Using the derived optimality conditions, an algorithm for solving a numerical model is developed and tested with the Miyun Reservoir in China.
Optimality Conditions for A Two-Stage Reservoir Operation Problem
NASA Astrophysics Data System (ADS)
Zhao, J.; Cai, X.; Wang, Z.
2010-12-01
This paper discusses the optimality conditions for standard operation policy (SOP) and hedging rule (HR) for a two-stage reservoir operation problem within a consistent theoretical framework. The effects of three typical constraints, which are mass balance, non-negative release and storage constraints under both certain and uncertain conditions have been analyzed. When all non-negative constraints and storage constraints are non-binding, HR results in optimal reservoir operation following the marginal benefit (MB) principle (the MB is equal over the two stages); while if any of the non-negative release or storage constraints is binding, in general SOP results in the optimal solution except two special cases. One of them is a complement of the traditional SOP/HR curve, which happens while the capacity constraint is binding; the other is a special hedging rule, which should be employed to carry over all water in the current stage to the future, when extreme drought is certain and higher marginal utility exists for the second stage. Furthermore, uncertainty complicates the effects of the various constraints but in general higher uncertainty level in the future makes HR a more favorable since water needs to be reserved to defense the risk caused by the uncertainty. Using the derived optimality conditions, an algorithm for solving the model numerically has been developed and tested with hypothetical examples.
New attitude penalty functions for spacecraft optimal control problems
Schaub, H.; Junkins, J.L.; Robinett, R.D.
1996-03-01
A solution of a spacecraft optimal control problem, whose cost function relies on an attitude description, usually depends on the choice of attitude coordinates used. A problem could be solved using 3-2-1 Euler angles or using classical Rodriguez parameters and yield two different ``optimal`` solutions, unless the performance index in invariant with respect to the attitude coordinate choice. Another problem arising with many attitude coordinates is that they have no sense of when a body has tumbled beyond 180{degrees} from the reference attitude. In many such cases it would be easier (i.e. cost less) to let the body complete the revolution than to force it to reverse the rotation and return to the desired attitude. This paper develops a universal attitude penalty function g() whose value is independent of the attitude coordinates chosen to represent it. Furthermore, this function will achieve its maximum value only when a principal rotation of {plus_minus}180{degrees} from the target state is performed. This will implicitly permit the g() function to sense the shortest rotational distance back to the reference state. An attitude penalty function which depends on the Modified Rodriguez Parameters (MRP) will also be presented. These recently discovered MRPs are a non-singular three-parameter set which can describe any three-attitude. This MRP penalty function is simpler than the attitude coordinate independent g() function, but retains the useful property of avoiding lengthy principal rotations of more than {plus_minus}180{degrees}.
Dillard, Amanda J; Midboe, Amanda M; Klein, William M P
2009-11-01
College students were identified who were unrealistically optimistic about the likelihood they would experience severe problems due to alcohol consumption. These individuals were then followed over a 2-year period to determine whether they were more likely to report experiencing a range of alcohol-related negative events. Unlike the majority of studies on unrealistic optimism, this study (a) assessed bias at the individual rather than group level and (b) used a prospective rather than cross-sectional design. Participants completed measures at four times, each separated by 4-6 months. Findings showed that unrealistic optimism at Time 1 was associated with a greater number of negative events at Times 2, 3, and 4. Similarly, unrealistic optimism at Time 2 was associated with more negative events at Times 3 and 4. In all cases, the relationships were significant when controlling for previous negative events, suggesting the effects of unrealistic optimism can mount over time. PMID:19721102
NASA Astrophysics Data System (ADS)
Rao, R. V.; Savsani, V. J.; Balic, J.
2012-12-01
An efficient optimization algorithm called teaching-learning-based optimization (TLBO) is proposed in this article to solve continuous unconstrained and constrained optimization problems. The proposed method is based on the effect of the influence of a teacher on the output of learners in a class. The basic philosophy of the method is explained in detail. The algorithm is tested on 25 different unconstrained benchmark functions and 35 constrained benchmark functions with different characteristics. For the constrained benchmark functions, TLBO is tested with different constraint handling techniques such as superiority of feasible solutions, self-adaptive penalty, ɛ-constraint, stochastic ranking and ensemble of constraints. The performance of the TLBO algorithm is compared with that of other optimization algorithms and the results show the better performance of the proposed algorithm.
Path Following in the Exact Penalty Method of Convex Programming
Zhou, Hua; Lange, Kenneth
2015-01-01
Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to ∞, one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and the solution is recovered for a finite value of the penalty constant. In practice, the kinks in the penalty and the unknown magnitude of the penalty constant prevent wide application of the exact penalty method in nonlinear programming. In this article, we examine a strategy of path following consistent with the exact penalty method. Instead of performing optimization at a single penalty constant, we trace the solution as a continuous function of the penalty constant. Thus, path following starts at the unconstrained solution and follows the solution path as the penalty constant increases. In the process, the solution path hits, slides along, and exits from the various constraints. For quadratic programming, the solution path is piecewise linear and takes large jumps from constraint to constraint. For a general convex program, the solution path is piecewise smooth, and path following operates by numerically solving an ordinary differential equation segment by segment. Our diverse applications to a) projection onto a convex set, b) nonnegative least squares, c) quadratically constrained quadratic programming, d) geometric programming, and e) semidefinite programming illustrate the mechanics and potential of path following. The final detour to image denoising demonstrates the relevance of path following to regularized estimation in inverse problems. In regularized estimation, one follows the solution path as the penalty constant decreases from a large value. PMID:26366044
Finite element solution of optimal control problems with inequality constraints
NASA Technical Reports Server (NTRS)
Bless, Robert R.; Hodges, Dewey H.
1990-01-01
A finite-element method based on a weak Hamiltonian form of the necessary conditions is summarized for optimal control problems. Very crude shape functions (so simple that element numerical quadrature is not necessary) can be used to develop an efficient procedure for obtaining candidate solutions (i.e., those which satisfy all the necessary conditions) even for highly nonlinear problems. An extension of the formulation allowing for discontinuities in the states and derivatives of the states is given. A theory that includes control inequality constraints is fully developed. An advanced launch vehicle (ALV) model is presented. The model involves staging and control constraints, thus demonstrating the full power of the weak formulation to date. Numerical results are presented along with total elapsed computer time required to obtain the results. The speed and accuracy in obtaining the results make this method a strong candidate for a real-time guidance algorithm.