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Solving Optimization Problems over the Weakly Efficient Set

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2021年第2期42卷 180-210页

作者： Sadeghi, Javad Mohebi, Hossein Kerman Grad Univ Adv Technol Dept Math Kerman Iran Shahid Bahonar Univ Kerman Dept Math POB 76169133 Kerman *** Iran Shahid Bahonar Univ Kerman Mahani Math Res Ctr POB 76169133 Kerman *** Iran

In this paper, we study the optimization problem (PWE) of minimizing a convex function over the set of weakly efficient solutions of a convex multiobjective problem. This is done by using the fact that each lower semicontinuous convex function is an upper envelope of its affine minorants together with a generalized cutting plane method. We give necessary conditions for optimal solutions of the problem (PWE). Moreover, a novel algorithm for solving the problem (PWE) together with numerical results are presented. We also prove that the proposed algorithm terminates after a finite numbers of iterations, and the algorithm is coded in MATLAB language and evaluated by numerical examples.

关键词： Nonlinear programming multiobjective programming nonconvex programming weak-efficiency generalized cutting plane method

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Truncated Nuclear Norm Minimization for Image Restoration Based on Iterative Support Detection

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MATHEMATICAL PROBLEMS IN ENGINEERING 2014年第1期2014卷 1-17页

作者： Wang, Yilun Su, Xinhua Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Peoples R China

Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing, and medical imaging, and these kinds of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the singular values are simultaneously minimized and the rank may not be well approximated (Hu et al., 2013). Correspondingly, in this paper, we propose a new multistage algorithm, which makes use of the concept of Truncated Nuclear Norm Regularization (TNNR) proposed by Hu et al., 2013, and iterative support detection (ISD) proposed by Wang and Yin, 2010, to overcome the above limitation. Besides matrix completion problems considered by Hu et al., 2013, the proposed method can be also extended to the general low-rank matrix recovery problems. Extensive experiments well validate the superiority of our new algorithms over other state-of-the-art methods.

关键词： IMAGE reconstruction IMAGE converters ITERATIVE methods (Mathematics) COMPRESSED sensing (Signal processing) nonconvex programming APPROXIMATION theory DIAGNOSTIC imaging

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Generating scenario trees for multistage decision problems

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MANAGEMENT SCIENCE 2001年第2期47卷 295-307页

作者： Hoyland, K Wallace, SW Norwegian Univ Sci & Technol Dept Ind Econ & Technol Management N-7491 Trondheim Norway

In models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models. If the uncertainties are expressed in terms of multivariate continuous distributions, or a discrete distribution with far too many outcomes, we normally face two possibilities: either creating a decision model with internal sampling, or trying to find a simple discrete approximation of the given distribution that serves as input to the model. This paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties. Users are free to specify any statistical properties they find relevant, and the method can handle inconsistencies in the specifications. The basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties. We illustrate the method by single- and multiple-period problems. The results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications. We discuss how to verify that the relevant statistical properties are captured in these specifications, and argue that what are the relevant properties, will be problem dependent.

关键词： scenario generation asset allocation nonconvex programming

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***: easy advanced global optimization in Julia

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OPTIMIZATION METHODS & SOFTWARE 2022年第2期37卷 425-450页

作者： Wilhelm, M. E. Stuber, M. D. Univ Connecticut Proc Syst & Operat Res Lab Dept Chem & Biomol Engn 191 Auditorium RdUnit 3222 Storrs CT 06269 USA

An extensible open-source deterministic global optimizer (EAGO) programmed entirely in the Julia language is presented. EAGO was developed to serve the need for supporting higher-complexity user-defined functions (e.g. functions defined implicitly via algorithms) within optimization models. EAGO embeds a first-of-its-kind implementation of McCormick arithmetic in an Evaluator structure allowing for the construction of convex/concave relaxations using a combination of source code transformation, multiple dispatch, and context-specific approaches. Utilities are included to parse user-defined functions into a directed acyclic graph representation and perform symbolic transformations enabling dramatically improved solution speed. EAGO is compatible with a wide variety of local optimizers, the most exhaustive library of transcendental functions, and allows for easy accessibility through the JuMP modelling language. Together with Julia's minimalist syntax and competitive speed, these powerful features make EAGO a versatile research platform enabling easy construction of novel meta-solvers, incorporation and utilization of new relaxations, and extension to advanced problem formulations encountered in engineering and operations research (e.g. multilevel problems, user-defined functions). The applicability and flexibility of this novel software is demonstrated on a diverse set of examples. Lastly, EAGO is demonstrated to perform comparably to state-of-the-art commercial optimizers on a benchmarking test set.

关键词： Deterministic global optimization nonconvex programming McCormick relaxations optimization software branch-and-bound Julia

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AN ALGORITHM FOR OPTIMIZING OVER THE WEALKY-EFFICIENT SET

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1986年第2期25卷 192-199页

作者： BENSON, HP UNIV FLORIDA DEPT MANAGEMENT & ADM SCIGAINESVILLEFL 32611 USA

In this paper a bisecting search algorithm is developed for solving the problem (P) of optimizing a linear function over the set of weakly-efficient solutions of a multiple objective linear program. We show that problem (P) can arise in a variety of practical situations. The algorithm for solving problem (P) is guaranteed to find an optimal or approximately-optimal solution for the problem in a finite number of steps. Using a FORTRAN computer code called CONMIN as an aid, we have solved ten test problems using our proposed algorithm. This preliminary computational experience seems to indicate that the algorithm is quite practical for relatively small problems. [ABSTRACT FROM AUTHOR]

关键词： efficiency Multiple objective linear programming nonconvex programming nonlinear programming walgorithms for nonlinear programming weak-efficiency

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Nonsmooth and nonconvex Optimization via Approximate Difference-of-Convex Decompositions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第1期182卷 49-80页

作者： van Ackooij, Wim de Oliveira, Welington Elect France EDF R&D Paris France PSL Res Univ CMA MINES ParisTech Sophia Antipolis France

We propose an optimization technique for computing stationary points of a broad class of nonsmooth and nonconvex programming problems. The proposed approach (approximately) decomposes the objective function as the difference of two convex functions and performs inexact optimization of the resulting (convex) subproblems. We prove global convergence of our method in the sense that, for an arbitrary starting point, every accumulation point of the sequence of iterates is a Clarke-stationary solution. The given approach is validated by encouraging numerical results on several nonsmooth and nonconvex distributionally robust optimization problems.

关键词： nonconvex programming Nonsmooth optimization Lower-C-2 functions DC decomposition

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An alternating structured trust region algorithm for separable optimization problems with nonconvex constraints

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2014年第2期57卷 365-386页

作者： Xue, Dan Sun, Wenyu Qi, Liqun Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210046 Jiangsu Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

In this paper, we propose a structured trust-region algorithm combining with filter technique to minimize the sum of two general functions with general constraints. Specifically, the new iterates are generated in the Gauss-Seidel type iterative procedure, whose sizes are controlled by a trust-region type parameter. The entries in the filter are a pair: one resulting from feasibility;the other resulting from optimality. The global convergence of the proposed algorithm is proved under some suitable assumptions. Some preliminary numerical results show that our algorithm is potentially efficient for solving general nonconvex optimization problems with separable structure.

关键词： nonconvex programming Trust region methods Alternating direction methods Separable structure Filter method

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OPTIMIZATION TECHNIQUE BASED ON LEARNING AUTOMATA

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第2期64卷 331-347页

作者： NAJIM, K PIBOULEAU, L LELANN, MV 1. Ecole Nationale Supérieure d'Ingénieurs de Génie Chimique Toulouse France

Optimization techniques are finding increasingly numerous applications in process design, in parallel to the increase of computer sophistication. The process synthesis problem can be stated as a largescale constrained optimization problem involving numerous local optima and presenting a nonlinear and nonconvex character. To solve this kind of problem, the classical optimization methods can lead to analytical and numerical difficulties. This paper describes the feasibility of an optimization technique based on learning systems which can take into consideration all the prior information concerning the process to be optimized and improve their behavior with time. This information generally occurs in a very complex analytical, empirical, or know-how form. Computer simulations related to chemical engineering problems (benzene chlorination, distillation sequence) and numerical examples are presented. The results illustrate both the performance and the implementation simplicity of this method.

关键词： Optimization learning systems nonconvex programming process synthesis

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HYPERSPECTRAL SUPER-RESOLUTION VIA LOW RANK TENSOR TRIPLE DECOMPOSITION

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2024年第3期20卷 942-966页

作者： Cui, Xiaofei Chang, Jingya Guangdong Univ Technol Sch Math & Stat Guangzhou Peoples R China

Hyperspectral image (HSI) and multispectral image (MSI) fusion aims at producing a super-resolution image (SRI). In this paper, we establish a nonconvex optimization model for image fusion problem through low rank tensor triple decomposition. Using the limited memory BFGS (L-BFGS) approach, we develop a first-order optimization algorithm for obtaining the desired super-resolution image (TTDSR). Furthermore, two detailed methods are provided for calculating the gradient of the objective function. With the aid of the KurdykaLojasiewicz property, the iterative sequence is proved to converge to a stationary point. Finally, experimental results on different datasets show the effectiveness of our proposed approach.

关键词： Key words and phrases. Tensor triple decomposition nonconvex programming Kurdyka-Lojasiewicz property image fusion hyperspectral image multispectral image

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A FINITE, NONADJACENT EXTREME-POINT SEARCH ALGORITHM FOR OPTIMIZATION OVER THE EFFICIENT SET

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1992年第1期73卷 47-64页

作者： BENSON, HP UNIV FLORIDA COLL BUSINESS ADMGAINESVILLEFL 32611 USA

The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.

关键词： MULTIPLE-CRITERIA DECISION MAKING EXTREME-POINT SEARCH GLOBAL OPTIMIZATION EFFICIENT SET nonconvex programming

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