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Constraint qualifications and KKT conditions for bilevel programming problems

MATHEMATICS OF OPERATIONS RESEARCH 2006年第4期31卷 811-824页

作者： Ye, Jane J. Univ Victoria Dept Math & Stat Victoria BC V8W 3P4 Canada

In this paper we consider the bilevel programming problem (BLPP), which is a sequence of two optimization problems where the constraint region of the upper-level problem is determined implicitly by the solution set to the lower-level problem. We extend well-known constraint qualifications for nonlinear programming problems such as the Abadie constraint qualification, the Kuhn-Tucker constraint qualification, the Zangwill constraint qualification, the Arrow-Hurwicz-Uzawa constraint qualification, and the weak reverse convex constraint qualification to BLPPs and derive a Karash-Kuhn-Tucker (KKT)-type necessary optimality condition under these constraint qualifications without assuming the lower-level problem satisfying the Mangasarian Fromovitz constraint qualification. Relationships among various constraint qualifications are also given.

关键词： necessary optimality conditions constraint qualifications nonsmooth analysis value function bilevel programming problems

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Co-evolutionary algorithm: An efficient approach for bilevel programming problems

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ENGINEERING OPTIMIZATION 2014年第3期46卷 361-376页

作者： Li, Hecheng Fang, Lei Qinghai Normal Univ Dept Math Minist Educ Key Lab Tibetan Informat Proc Xining Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing Peoples R China Nankai Univ Sch Business Tianjin 300071 Peoples R China

The bilevel programming problem involves two optimization problems, which is hierarchical, strongly NP-hard and very challenging for most existing optimization approaches. An efficient universal co-evolutionary algorithm is developed in this article to deal with various bilevel programming problems. In the proposed algorithm, evolutionary algorithms are used to explore the leader's and the follower's decision-making spaces interactively. Unlike other existing approaches, in the suggested procedure the follower's problem is solved in two phases. First, an evolutionary algorithm is run for a few generations to obtain an approximation of lower level solutions. In the second phase, from all approximate solutions obtained above, only a small number of good points are selected and evolved again by a newly designed multi-criteria evolutionary algorithm. The technique refines some candidate solutions and can efficiently reduce the computational cost of obtaining feasible solutions. Proof-of-principle experiments demonstrate the efficiency of the proposed approach.

关键词： co-evolutionary algorithm bilevel programming problems approximate solutions optimal solutions

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Exact penalty functions for convex bilevel programming problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2001年第3期110卷 621-643页

作者： Liu, GS Han, JY Zhang, JZ Renmin Univ China Sch Business Adm Beijing Peoples R China Chinese Acad Sci Inst Appl Math Beijing Peoples R China City Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

In this paper, we propose a new constraint qualification for convex bilevel programming problems. Under this constraint qualification, a locally and globally exact penalty function of order I for a single-level reformulation of convex bilevel programming problems is given without requiring the linear independence condition and the strict complementarity condition to hold in the lower-level problem. Based on these results, locally and globally exact penalty functions for two other single-level reformulations of convex bilevel programming problems can be obtained. Furthermore, sufficient conditions for partial calmness to hold in some single-level reformulations of convex bilevel programming problems can be given.

关键词： bilevel programming problems constraint qualifications exact penalty functions reformulations partial calmness

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Stability of regularized bilevel programming problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1997年第3期93卷 575-596页

作者： Lignola, MB Morgan, J UNIV NAPLES FEDERICO II FAC SCI DIPARTIMENTO MATEMAT & APPLICAZ NAPLES ITALY

A bilevel programming problem S is considered. First, sufficient conditions of minimal character are given on the data of the problem in order to guarantee the lower semicontinuity of the marginal function of the upper level problem. Then, for epsilon > 0, a regularized problem S(epsilon) is considered for which continuity of the regularized marginal function and convergence of the approximate value, as epsilon goes to zero, are obtained. Moreover, under perturbations on the data, convergence results for the perturbed marginal functions and the solutions to the problem S-n(epsilon) are given for any epsilon > 0.

关键词： bilevel programming problems strong Stackelberg problems marginal functions epiconvergence convergence of multifunctions convergence of solutions convergence of values

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Optimality conditions for bilevel programming problems

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Optimization 1995年第1期33卷 9-27页

作者： Ye, J.J. Zhu, D.L. Department of Mathematics and Statistics University of Victoria Victoria B.C V8W 3P4 Canada GERAD École des Hautes Études Commerciales ontréal Québec H3 T 1V6 5255 Avenur decelles Canada

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementarity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also given for the case where the lower level problem is unconstrained. © 1995, Taylor & Francis Group, LLC. All rights reserved.

关键词： bilevel programming problems Constraint qualification Nonsmooth analysis Optimality conditions

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A genetic algorithm for solving a special class of Nonlinear bilevel programming problems

A genetic algorithm for solving a special class of Nonlinear...

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7th International Conference on Computational Science (ICCS 2007)

作者： Li, Hecheng Wang, Yuping Xidian Univ Sch Comp Sci & Technol Xian 710071 Peoples R China Xidian Univ Sch Sci Xian 710071 Peoples R China

ISBN: (纸本)9783540725893

A special nonlinear bilevel programming problem (BLPP), whose follower-level problem is a convex programming with a linear objective function in y, is transformed into an equivalent single-level programming by using Karush-Kuhn-Tucker (K-K-T) conditions. To solve the equivalent problem effectively, a new genetic algorithm is proposed. First, a linear programming (LP) is constructed to decrease the dimensions of the transformed problem. Then based on a constraint-handling scheme, a second-phase evolving process is designed for some offspring of crossover and mutation, in which the linear property of follower's function is used to generate high quality potential offspring.

关键词： bilevel programming problems genetic algorithm linear programming problem constraint handling optimal solutions

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A new descent method for solving ill-posed bilevel programming problems via Maxmin model

A new descent method for solving ill-posed Bilevel programmi...

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4th International Conference on Digital Manufacturing and Automation (ICDMA)

作者： Jia Shihui Wuhan Univ Sci & Technol Sch Sci Dept Math Wuhan 430081 Peoples R China

ISBN: (纸本)9780769550169

The results of optimistic model and pessimistic model are always not efficient for practical application because they are two extreme possibilities for ill-posed bilevel programming problem. This paper presents a new model by transferring Minmax model to a Maxmin model to solve above shortcoming. And we prove, by this transformation, a more practical value between the optimistic value and the pessimistic value can be achieved. Then, a new descent algorithm is developed to solve this new model by considering the satisfying degree of the lower level. Finally, an illustrated numerical example is demonstrated to show the proposed new model and algorithm are feasible and we can get a better result from all the iterated points by this Maxmin model than by the two extreme models.

关键词： bilevel programming problems Maxmin problem Satisfying degree Descent algorithm

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Duality for Multiobjective bilevel programming problems with Extremal-Value Function

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Journal of Mathematical Research with Applications 2015年第3期35卷 311-320页

作者： Haijun WANG Ruifang ZHANG Department of Mathematics Taiyuan Normal University

For a multiobjective bilevel programming problem(P) with an extremal-value function,its dual problem is constructed by using the Fenchel-Moreau conjugate of the functions *** some convexity and monotonicity assumption... 详细信息

关键词： multiobjective optimization bilevel programming problems conjugate duality convex programming composed convex functions

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Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution

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JOURNAL OF GLOBAL OPTIMIZATION 2000年第1期16卷 57-67页

作者： Lignola, MB Morgan, J Univ Naples Federico II Dipartimento Matemat & Applicaz R Caccioppoli I-80125 Naples Italy Compl Univ Monte S Angelo I-80126 Naples Italy

We introduce various notions of well-posedness for a family of variational inequalities and for an optimization problem with constraints defined by variational inequalities having a unique solution. Then, we give suff... 详细信息

关键词： optimization problems with variational inequalities constraints bilevel programming problems Stackelberg problems approximating sequences strong well-posedness generalized well-posedness gap function monotone operators

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Sensitivity analysis of the value function for optimization problems with variational inequality constraints

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2001年第3期40卷 699-723页

作者： Lucet, Y Ye, JJ Simon Fraser Univ Ctr Expt & Construct Math Burnaby BC V5A 1S6 Canada Univ Victoria Dept Math & Stat Victoria BC V8W 3P4 Canada

In this paper we perform sensitivity analysis for optimization problems with variational inequality constraints (OPVICs). We provide upper estimates for the limiting subdifferential (singular limiting subdifferential) of the value function in terms of the set of normal (abnormal) coderivative (CD) multipliers for OPVICs. For the case of optimization problems with complementarity constraints (OPCCs), we provide upper estimates for the limiting subdifferentials in terms of various multipliers. An example shows that the other multipliers may not provide useful information on the subdifferentials of the value function, while the CD multipliers may provide tighter bounds. Applications to sensitivity analysis of bilevel programming problems are also given.

关键词： sensitivity analysis optimization problems variational inequality constraints complementarity constraints limiting subdifferentials value functions bilevel programming problems

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