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Solution of a general linear complementarity problem using smooth optimization and its application to bilinear programming and LCP

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APPLIED MATHEMATICS AND OPTIMIZATION 2001年第1期43卷 1-19页

作者： Fernandes, L Friedlander, A Guedes, M Júdice, J Escola Super Technol Tomar P-2300 Tomar Portugal Univ Estadual Campinas Dept Matemat Aplicada BR-13081 Campinas Brazil Univ Porto Fac Ciencias Dept Matemat Aplicada P-4000 Oporto Portugal Univ Coimbra Dept Matemat P-3000 Coimbra Portugal

This paper addresses a General Linear Complementarity Problem (GLCP) that has found applications in global optimization. It is shown that a solution of the GLCP can be computed by finding a stationary point of a differentiable function over a set defined by simple bounds on the variables. The application of this result to the solution of bilinear programs and LCPs is discussed. Some computational evidence of its usefulness is included in the last part of the paper.

关键词： global optimization linear complementarity problems bilinear programming box constrained optimization

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PSO-based and SA-based metaheuristics for bilinear programming problems: an application to the pooling problem

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JOURNAL OF HEURISTICS 2016年第2期22卷 147-179页

作者： Erbeyoglu, Gokalp Bilge, Umit Bogazici Univ Dept Ind Engn TR-34342 Istanbul Turkey

bilinear programming problems (BLP) are subsets of nonconvex quadratic programs and can be classified as strongly NP-Hard. The exact methods to solve the BLPs are inefficient for large instances and only a few heuristic methods exist. In this study, we propose two metaheuristic methods, one is based on particle swarm optimization (PSO) and the other is based on simulated annealing (SA). Both of the proposed approaches take advantage of the bilinear structure of the problem. For the PSO-based method, a search variable, which is selected among the variable sets causing bilinearity, is subjected to particle swarm optimization. The SA-based procedure incorporates a variable neighborhood scheme. The pooling problem, which has several application areas in chemical industry and formulated as a BLP, is selected as a test bed to analyze the performances. Extensive experiments are conducted and they indicate the success of the proposed solution methods.

关键词： bilinear programming Metaheuristics Pooling problem Particle swarm optimization Simulated annealing

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Modified difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers

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JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2015年第4期29卷 1607-1618页

作者： Verma, Tina Kumar, Amit Appadoo, S. S. Thapar Univ Sch Math & Comp Applicat Patiala 147004 Punjab India Univ Manitoba Dept Supply & Chain Management Winnipeg MB R3T 2N2 Canada

Li and Yang [D.F. Li and J. Yang, A difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers, Journal of Applied Mathematics 2013 (2013), 1-10] pointed out that there is no method in the literature for solving such bimatrix games in which payoffs are represented by intuitionistic fuzzy numbers and proposed a method for the same. In this paper, it is pointed out that Li and Yang have considered some mathematical incorrect assumptions in their proposed method. For resolving the shortcomings of Li and Yang's method, a new method (named as Mehar method) is proposed. Also, the exact optimal solution of the numerical problem, solved by Li and Yang by their proposed method, is obtained by the proposed Mehar method.

关键词： Bimatrix game trapezoidal intuitionistic fuzzy numbers difference-index based ranking bilinear programming

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Bidimensional packing by bilinear programming

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MATHEMATICAL programming 2009年第1期118卷 75-108页

作者： Caprara, Alberto Monaci, Michele Univ Bologna DEIS I-40136 Bologna Italy Univ Padua DEI I-35131 Padua Italy

We consider geometric problems in which rectangles have to be packed into (identical) squares. These problems turn out to be very hard in practice, and ILP formulations in which variables specify the coordinates in the packing perform very poorly. While most methods developed so far are based on simple geometric considerations, a recent landmark result of Fekete and Schepers suggests to put these geometric aspects aside and use the most advanced tools for the one-dimensional case. In this paper we make additional progress in this direction, especially on the basic question "Does a given set of rectangles fit into a square?", which turns out to be the bottleneck of all the approaches known. Given a set of rectangles and the associated convex hull of rectangle subsets that fit into a square, we derive a wide class of valid inequalities for this convex hull from a complete description of the two knapsack polytopes associated with the widths and the heights of the rectangles, respectively. In addition, we illustrate how to solve the associated separation problem as a bilinear program, for which we develop a solution method that turns out to be fast in practice, and show that the integer solutions that satisfy all these constraints generally correspond to vertices of the original convex hull for the benchmark instances in the literature. The same tools are used to derive lower bounds for the two-dimensional bin packing problem, corresponding to the determination of an optimal pair of so-called dual feasible functions. These lower bounds in many cases equal those obtained by the customary set covering formulation (for which column generation is very hard), but are computable within a time that is smaller by some orders of magnitude. This allows us to close a few of the benchmark instances in the literature.

关键词： Bidimensional packing bilinear programming Dual feasible functions Computational experiments

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Numerical Solution of bilinear programming Problems

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2008年第2期48卷 225-241页

作者： Orlov, A. V. Russian Acad Sci Inst Syst Dynam & Control Theory Siberian Branch Irkutsk 664033 Russia

A bilinear programming problem with uncoupled variables is considered. First, a special technique for generating test bilinear problems is considered. Approximate algorithms for local and global search are proposed. Asymptotic convergence of these algorithms is analyzed, and stopping rules are proposed. In conclusion, numerical results for randomly generated bilinear problems are presented and analyzed.

关键词： nonconvex optimization bilinear programming test problem generation local search critical point stopping rule global search numerical experiment

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On the Design of Constrained PI-Like Output Feedback Tracking Controllers via Robust Positive Invariance and bilinear programming

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IEEE CONTROL SYSTEMS LETTERS 2023年 7卷 1429-1434页

作者： dos Santos, Geovana Franca Castelan, Eugenio B. Lucia, Walter Concordia Univ CIISE Dept Montreal PQ Canada Univ Fed Santa Catarina Dept Automation & Syst Engn BR-88040900 Florianopolis Brazil

In this letter, we design a novel PI-like output feedback tracking controller for linear systems subject to state and input constraints. The proposed solution exploits the internal model principle to design via robust positive invariance and the Extended Farkas' Lemma, a constrained set-point tracking controller. The considered polyhedral setup results in a bilinear optimization design problem that can be solved offline, allowing to simultaneously determine the sets of admissible set-points and plant's initial conditions. The proposed solution guarantees asymptotic offset-free set-point tracking, local stability, and constraints fulfillment. Simulation results show the effectiveness of the proposed design and contrast it with an alternative scheme.

关键词： Linear systems Stability criteria programming PI control Optimization Economic indicators Asymptotic stability bilinear programming constrained set-point tracking PI-Control robust positive invariance

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A FINITELY CONVERGENT ALGORITHM FOR bilinear programming-PROBLEMS USING POLAR CUTS AND DISJUNCTIVE FACE CUTS

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MATHEMATICAL programming 1980年第1期19卷 14-31页

作者： SHERALI, HD SHETTY, CM Georgia Institute of Technology School of Industrial and Systems Engineering Atlanta GA USA

A finite algorithm is presented in this study for solving bilinear programs. This is accomplished by developing a suitable cutting plane which deletes at least a face of a polyhedral set. At an extreme point, a polar cut using negative edge extensions is used. At other points, disjunctive cuts are adopted. Computational experience on test problems in the literature is provided.

关键词： bilinear programming Polar Cuts Disjunctive Cuts Cutting Plane

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GENERALIZED bilinear-programming .1. MODELS, APPLICATIONS AND LINEAR-programming RELAXATION

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1992年第3期60卷 306-314页

作者： ALKHAYYAL, FA GEORGIA INST TECHNOL SCH IND & SYST ENGNATLANTAGA 30332 USA

The evolution of the bilinear programming problem is reviewed and a new, more general model is discussed. The model involves two decision vectors and reduces to a linear program when one of the decision vectors is fixed. The class of problems under consideration contains traditional bilinear programs, general quadratic programs, and bilinearly constrained and quadratically constrained extensions of these problems. We describe how several important applications from the literature, including the multiple modular design problem, can be modeled as generalized bilinear programs. Finally, we derive a linear programming relaxation that can be used as a subproblem in algorithmic solution schemes based on outer approximation and branch and bound.

关键词： bilinear programming QUADRATIC programming NONCONVEX programming MODULAR DESIGN PROBLEM

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Generalized bilinear programming: An application in farm management

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1996年第1期90卷 102-114页

作者： BloemhofRuwaard, JM Hendrix, EMT AGR UNIV WAGENINGEN DEPT MATH 6703 HA WAGENINGEN NETHERLANDS AGR UNIV WAGENINGEN CTR ENVIRONM & CLIMATE STUDIES 6700 HB WAGENINGEN NETHERLANDS

In this paper we analyze a model which arises from considering manure and fertilizer aspects in the farm management modelling for cattle husbandry. Modelling these aspects may lead to a multi-extremal problem with bilinear terms. Solution techniques for this problem are discussed and a specific branch and bound procedure is outlined. An example illustrates the multi-extremal structure of the problem and the solution methods mentioned.

关键词： agriculture environment branch and bound bilinear programming

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Efficient algorithms for solving rank two and rank three bilinear programming problems

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Journal of Global Optimization 1991年第2期1卷 155-171页

作者： Yajima, Yasutoshi Konno, Hiroshi Department of Industrial Engineering and Management Tokyo Institute of Technology Japan Institute of Human and Social Sciences Tokyo Institute of Technology Japan

Finitely convergent algorithms for solving rank two and three bilinear programming problems are proposed. A rank k bilinear programming problem is a nonconvex quadratic programming problem with the following structure: {Mathematical expression} where X ⊂ Rⁿ¹ and Y ⊂ Rⁿ² are non-empty and bounded polytopes. We show that a variant of parametric simplex algorithm can solve large scale rank two bilinear programming problems efficiently. Also, we show that a cutting-cake algorithm, a more elaborate variant of parametric simplex algorithm can solve medium scale rank three problems. © 1991 Kluwer Academic Publishers.

关键词： bilinear programming global minimization nonconvex quadratic programming problem parametric simplex algorithm

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