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13
A Smoothing Method For Mathematical Programs With Equilibrium Constraints
, 1996
"... The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalitie ..."
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Cited by 65 (6 self)
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The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalities are considered in this paper. They are transformed into an equivalent onelevel nonsmooth optimization problem. Then, a sequence of smooth, regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced. It is shown that the solutions (stationary points) of the approximate problems converge to a solution (stationary point) of the original MPEC problem. Numerical results showing viability of the approach are reported.
QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints
"... . We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) co ..."
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Cited by 24 (8 self)
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. We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, firstlevel constraints are linear, and secondlevel (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, illconditioning, convexity of the objective, monotonicity and symmetry of the secondlevel problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.maths.mu.OZ.AU/~danny/qpecgendoc.h...
Generalized stationary points and an interiorpoint method for mathematical programs with equilibrium constraints
 Industrial Engineering & Management Sciences, Northwestern University
, 2005
"... Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier proble ..."
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Cited by 15 (1 self)
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Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPECLICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interiorpoint algorithm failed to find a stationary point. Key words: Global convergence, interiorpoint methods, mathematical programming with equilibrium constraints, stationary point
A DecompositionBased Global Optimization Approach for Solving Bilevel Linear and Quadratic Programs
 in C. A. Floudas eds., State of the Art in Global Optimization
, 1996
"... . The paper presents a decomposition based global optimization approach to bilevel linear and quadratic programming problems. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed to a single yet nonconvex, due to the complementarity condition, ma ..."
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Cited by 6 (0 self)
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. The paper presents a decomposition based global optimization approach to bilevel linear and quadratic programming problems. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed to a single yet nonconvex, due to the complementarity condition, mathematical program. Based on the primaldual global optimization approach of Floudas and Visweswaran (1990, 1993), the problem is decomposed into a series of primal and relaxeddual subproblemswhose solutions provide lower and upper bounds to the global optimum. By further exploiting the special structure of the bilevel problem, new properties are established which enable the efficient implementation of the proposed algorithm. Computational results are reported for both linear and quadratic example problems. 1. Introduction Bilevel programming refers to optimization problems in which the constraint region is implicitly determined by another optimization problem, as follows: min x F (x; y)...
A TrustRegion Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience
, 2005
"... We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upperlevel objective and all constraintdefining functions, as well as a quadratic approximation of the lowerlevel objective ..."
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Cited by 4 (1 self)
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We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upperlevel objective and all constraintdefining functions, as well as a quadratic approximation of the lowerlevel objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.
Successive Convex Relaxation Approach to Bilevel Quadratic Optimization Problems
 Dept
, 1999
"... . The quadratic bilevel programming problem is an instance of a quadratic hierarchical decision process where the lower level constraint set is dependent on decisions taken at the upper level. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed t ..."
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. The quadratic bilevel programming problem is an instance of a quadratic hierarchical decision process where the lower level constraint set is dependent on decisions taken at the upper level. By replacing the inner problem by its corresponding KKT optimality conditions, the problem is transformed to a single yet nonconvex quadratic program, due to the complementarity condition. In this paper we adopt the successive convex relaxation approach proposed by Kojima and Tun¸cel for computing a convex relaxation of a nonconvex feasible region. By further exploiting the special structure of the bilevel problem, we establish new techniques which enable the efficient implementation of the proposed algorithm. The performance of these techniques is tested in a comparison with other procedures using a number of test problems of quadratic bilevel programming. 1 Introduction. Bilevel programming (abbreviated by BP) belongs to a class of nonconvex global optimization problems. It arises where decis...
Feature Article: The Definition of Optimal Solution and an Extended KunhnTucker Approach … 1 The Definition of Optimal Solution and an Extended KuhnTucker Approach for Fuzzy Linear Bilevel Programming
"... Abstract — Bilevel decision techniques are mainly developed for solving decentralized management problems with decision makers in a hierarchical organization. Organizational bilevel decisionmaking, such as planning of landuse, transportation and water resource, all may involve uncertain factors. T ..."
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Abstract — Bilevel decision techniques are mainly developed for solving decentralized management problems with decision makers in a hierarchical organization. Organizational bilevel decisionmaking, such as planning of landuse, transportation and water resource, all may involve uncertain factors. The parameters shown in a bileved programming model, either in the objective functions or constraints, are thus often imprecise, which is called fuzzy parameter bilevel programming (FPBLP) problem. Following our previous work [1, 2], this study first proposes a model of FPBLP. It then gives the definition of optimal solution for an FPBLP problem. Based on the definition and related theorems, this study develops a fuzzy number based KuhnTucher approach to solve the proposed FPBLP problem. Finally, an example further illustrates the power of the fuzzy number based KuhnTucher approach. Index Terms — Linear bilevel programming, KuhnTucker approach, Fuzzy set, Optimization.
DOI: 10.1007/s1028800500710 Bilevel programming: A survey
"... Abstract. This paper provides an introductory survey of a class of optimization problems known as bilevel programming. We motivate this class through a simple application, and then proceed with the general formulation of bilevel programs. We consider various cases (linear, linearquadratic, nonlinea ..."
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Abstract. This paper provides an introductory survey of a class of optimization problems known as bilevel programming. We motivate this class through a simple application, and then proceed with the general formulation of bilevel programs. We consider various cases (linear, linearquadratic, nonlinear), describe their main properties and give an overview of solution approaches.
DOI 10.1007/s1047900701762 An overview of bilevel optimization
, 2007
"... Abstract This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connect ..."
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Abstract This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).
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"... QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints∗ ..."
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QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints∗