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A MATRIX-FREE ALGORITHM FOR EQUALITY CONSTRAINED OPTIMIZATION PROBLEMS WITH RANK-DEFICIENT JACOBIANS

SIAM JOURNAL ON OPTIMIZATION 2009年第3期20卷 1224-1249页

作者： Curtis, Frank E. Nocedal, Jorge Waechter, Andreas NYU Courant Inst Math Sci New York NY 10003 USA Northwestern Univ Dept Elect Engn & Comp Sci Evanston IL 60208 USA IBM TJ Watson Res Ctr Yorktown Hts NY 10598 USA

We present a line search algorithm for large-scale constrained optimization that is robust and efficient even for problems with (nearly) rank-deficient Jacobian matrices. The method is matrix-free (i.e., it does not require explicit storage or factorizations of derivative matrices), allows for inexact step computations, and is applicable for nonconvex problems. The main components of the approach are a trust region subproblem for handling ill-conditioned or inconsistent linear models of the constraints and a process for attaining a sufficient reduction in a local model of a penalty function. We show that the algorithm is globally convergent to first-order optimal points or to stationary points of an infeasibility measure. Numerical results are presented.

关键词： large-scale optimization constrained optimization nonconvex programming trust regions inexact linear system solvers Krylov subspace methods

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nonconvex sensitivity-based generalized Benders decomposition

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JOURNAL OF GLOBAL OPTIMIZATION 2023年第1期86卷 37-60页

作者： Lin, Jia-Jiang Xu, Feng Luo, Xiong-Lin China Univ Petr Dept Automat Beijing 102249 Peoples R China

This paper considers general separable pseudoconvex optimization problems with continuous complicating variables in which primal and projected problems are both pseudoconvex problems. A novel decomposition method based on generalized Benders decomposition, named nonconvex sensitivity-based generalized Benders decomposition, is developed and proved strictly to obtain optimal solutions of general separable pseudoconvex optimization problems of interest without constructing surrogate models. By the use of a reformulation strategy (introducing an extra equality constraint and constructing several subproblems), the algorithm handles the nonconvexity by direct manipulations of consistent linear Benders cuts and the check of optimality conditions and approximating the feasible region of complicating variables by supporting hyperplanes. The master problems of the new algorithm are always linear programming problems and the solution of the algorithm contains sensitivity information about complicating variables. Moreover, the new algorithm could also be used as a tool to check the nonconvexity of an optimization problem. Two cases are given to confirm the validity and applicability of the proposed algorithm.

关键词： nonconvex programming Separable pseudoconvexity Benders decomposition Optimality conditions Sensitivity analysis

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JOINTLY CONSTRAINED BICONVEX programming

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MATHEMATICS OF OPERATIONS RESEARCH 1983年第2期8卷 273-286页

作者： ALKHAYYAL, FA FALK, JE GEORGE WASHINGTON UNIV DEPT OPERAT RESWASHINGTONDC 20052

This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x and y over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a special case of the problem addressed in this paper. We prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point. The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented.

关键词： biconvex functions. Bilinear programming branch and bound algorithm global convergence nonconvex programming

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Integer programming Formulations for Approximate Packing Circles in a Rectangular Container

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MATHEMATICAL PROBLEMS IN ENGINEERING 2014年第1期2014卷 1-6页

作者： Litvinchev, Igor Ozuna Espinosa, Edith Lucero Russian Acad Sci Ctr Comp Complex Syst Dept Moscow 119333 Russia Nuevo Leon State Univ Fac Mech & Elect Engn Monterrey 66450 NL Mexico

A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace, and food industries. Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. New formulations are proposed for approximate solution of packing problem. The container is approximated by a regular grid and the nodes of the grid are considered as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. Nesting circles inside one another is also considered. The resulting binary problem is then solved by commercial software. Numerical results are presented to demonstrate the efficiency of the proposed approach and compared with known results.

关键词： INTEGER programming CONTAINERS LOGISTICS HEURISTIC algorithms nonconvex programming MATHEMATICAL optimization

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Efficient Method for Flexibility Analysis of Large-Scale nonconvex Heat Exchanger Networks

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INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH 2015年第43期54卷 10757-10767页

作者： Li, Jilong Du, Jian Zhao, Zongchang Yao, Pingjing Dalian Univ Technol Dept Chem Engn State Key Lab Fine Chem Dalian 116024 Liaoning Peoples R China Chinese Acad Sci Shenyang Inst Automat Shenyang 110016 Liaoning Peoples R China

A novel method is proposed for flexibility analysis of nonconvex heat exchanger networks. In this method, the direction matrix is introduced to describe the deviation direction of the uncertain parameter. Then by searching the critical directions that restrict the process flexibility with the simulated annealing algorithm, the flexibility index can be obtained. Since the directions are no longer limited to the vertices, this method can well deal with the nonconvex problems. Moreover, for large-scale problems, a decoupling strategy is developed to enhance the efficiency. On the basis of this, the entire network is decomposed into several independent subnetworks, and the one with the lowest flexibility decides the global flexibility index. Two examples are studied, and the results well demonstrate the various aspects dealt with in this work.

关键词： FLEXIBILITY (Mechanics) HEAT exchangers SIMULATED annealing (Mathematics) DECOMPOSITION (Chemistry) nonconvex programming

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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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***: 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 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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