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27
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 52 (5 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.
Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints
 SIAM Journal on Optimization
, 1997
"... Mathematical programs with nonlinear complementarity constraints are reformulated using betterposed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the ex ..."
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Cited by 35 (0 self)
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Mathematical programs with nonlinear complementarity constraints are reformulated using betterposed but nonsmooth constraints. We introduce a class of functions, parameterized by a real scalar, to approximate these nonsmooth problems by smooth nonlinear programs. This smoothing procedure has the extra benefits that it often improves the prospect of feasibility and stability of the constraints of the associated nonlinear programs and their quadratic approximations. We present two globally convergent algorithms based on sequential quadratic programming, SQP, as applied in exact penalty methods for nonlinear programs. Global convergence of the implicit smooth SQP method depends on existence of a lowerlevel nondegenerate (strictly complementary) limit point of the iteration sequence. Global convergence of the explicit smooth SQP method depends on a weaker property, i.e. existence of a limit point at which a generalized constraint qualification holds. We also discuss some practical matter...
Stochastic Mathematical Programs With Equilibrium Constraints
 Operations Research Letters
, 1997
"... We introduce Stochastic Mathematical Programs with Equilibrium Constraints (SMPEC), which generalize MPEC models to explicitly incorporate possible uncertainties in the problem data to obtain robust solutions to hierarchical problems. For this problem, we establish results on the existence of soluti ..."
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Cited by 24 (5 self)
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We introduce Stochastic Mathematical Programs with Equilibrium Constraints (SMPEC), which generalize MPEC models to explicitly incorporate possible uncertainties in the problem data to obtain robust solutions to hierarchical problems. For this problem, we establish results on the existence of solutions, and on the convexity and directional differentiability of the implicit upperlevel objective function, both for continuously and discretely distributed probability distributions. In so doing, we establish the links between SMPEC models and twostage stochastic programs with recourse. We also introduce basic parallel iterative algorithms for discretely distributed SMPEC problems. 1 Introduction The present paper serves to introduce a framework for hierarchical decisionmaking under uncertainty. Hierarchical decisionmaking problems are encountered in a wide variety of domains in the engineering and experimental natural sciences, and in regional planning, management, and economics. These ...
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 20 (5 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...
Some Feasibility Issues in Mathematical Programs with Equilibrium Constraints
"... This paper is concerned with some feasibility issues in mathematical programs with equilibrium constraints (MPECs) where additional joint constraints are present that must be satisfied by the state and design variables of the problems. We introduce sufficient conditions that guarantee the feasibilit ..."
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Cited by 18 (7 self)
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This paper is concerned with some feasibility issues in mathematical programs with equilibrium constraints (MPECs) where additional joint constraints are present that must be satisfied by the state and design variables of the problems. We introduce sufficient conditions that guarantee the feasibility of these MPECs. It turns out that these conditions also guarantee the feasibility of the quadratic programming subproblems arising from the penalty interior point algorithm (PIPA) and the sequential quadratic programming (SQP) algorithm for solving MPECs; thus the same conditions ensure that these algorithms are applicable for solving this class of jointly constrained MPECs.
Complementarity Problems in GAMS and the PATH Solver
 Journal of Economic Dynamics and Control
, 1998
"... A fundamental mathematical problem is to find a solution to a square system of nonlinear equations. There are many methods to approach this problem, the most famous of which is Newton's method. In this paper, we describe a generalization of this problem, the complementarity problem. We show how such ..."
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Cited by 17 (6 self)
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A fundamental mathematical problem is to find a solution to a square system of nonlinear equations. There are many methods to approach this problem, the most famous of which is Newton's method. In this paper, we describe a generalization of this problem, the complementarity problem. We show how such problems are modeled within the GAMS modeling language and provide details about the PATH solver, a generalization of Newton's method, for finding a solution. While the modeling format is applicable in many disciplines, we draw the examples in this paper from an economic background. Finally, some extensions of the modeling format and the solver are described. Keywords: Complementarity problems, variational inequalities, algorithms AMS Classification: 90C33,65K10 This paper is an extended version of a talk presented at CEFES '98 (Computation in Economics, Finance and Engineering: Economic Systems) in Cambridge, England in July 1998 This material is based on research supported by Nationa...
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 (0 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
Complementarity And Related Problems: A Survey
, 1998
"... This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to ..."
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Cited by 14 (0 self)
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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.
Piecewise Sequential Quadratic Programming For Mathematical Programs With . . .
"... We describe some first and secondorder optimality conditions for mathematical programs with equilibrium constraints (MPEC). Mathematical programs with parametric nonlinear complementarity constraints are the focus. Of interest is the result that under a linear independence assumption that is stand ..."
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Cited by 12 (5 self)
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We describe some first and secondorder optimality conditions for mathematical programs with equilibrium constraints (MPEC). Mathematical programs with parametric nonlinear complementarity constraints are the focus. Of interest is the result that under a linear independence assumption that is standard in nonlinear programming, the otherwise combinatorial problem of checking whether a point is stationary for an MPEC is reduced to checking stationarity of single nonlinear program. We also present a piecewise sequential quadratic programming (PSQP) algorithm for solving MPEC. Local quadratic convergence is shown under the linear independence assumption and a secondorder sufficient condition. Some computational results are given. KEY WORDS MPEC, bilevel program, nonlinear complementarity problem, nonlinear program, first and secondorder optimality conditions, linear independence constraint qualification, sequential quadratic programming, quadratic convergence. 2 Chapter 1 1 INTRODUC...
Traffic Modeling and Variational Inequalities using GAMS
 Operations Research and Decision Aid Methodologies in Traffic and Transportation Management, NATO ASI Series F
, 1997
"... We describe how several traffic assignment and design problems can be formulated within the GAMS modeling language using newly developed modeling and interface tools. The fundamental problem is user equilibrium, where multiple drivers compete noncooperatively for the resources of the traffic network ..."
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Cited by 10 (3 self)
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We describe how several traffic assignment and design problems can be formulated within the GAMS modeling language using newly developed modeling and interface tools. The fundamental problem is user equilibrium, where multiple drivers compete noncooperatively for the resources of the traffic network. A description of how these models can be written as complementarity problems, variational inequalities, mathematical programs with equilibrium constraints, or stochastic linear programs is given. At least one general purpose solution technique for each model format is briefly outlined. Some observations relating to particular model solutions are drawn. 1 Introduction Models that postulate ways to assign traffic within a transportation network for a given demand have been used in planning and analysis for many years, see [41] and references therein. A popular technique for such assignment is to use the shortest path between the origin and destination points of a given journey. Of course, s...