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12
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.
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...
Exact penalization and necessary optimality conditions for generalized bilevel programming problems
 SIAM J. Optim
, 1997
"... Abstract. The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the ..."
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Cited by 17 (15 self)
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Abstract. The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform parametric error bounds as penalty functions gives single level problems equivalent to the GBLP. Several local and global uniform parametric error bounds are presented, and assumptions guaranteeing that they apply are discussed. We then derive Kuhn–Tuckertype necessary optimality conditions by using exact penalty formulations and nonsmooth analysis. Key words. generalized bilevel programming problems, variational inequalities, exact penalty formulations, uniform parametric error bounds, necessary optimality conditions, nonsmooth analysis
Exact And Inexact Penalty Methods For The Generalized Bilevel Programming Problem
 Mathematical Programming
, 1996
"... We consider a hierarchical system where a leader incorporates into its strategy the reaction of the follower to its decision. The follower's reaction is quite generally represented as the solution set to a monotone variational inequality. For the solution of this nonconvex mathematical program a pen ..."
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Cited by 15 (3 self)
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We consider a hierarchical system where a leader incorporates into its strategy the reaction of the follower to its decision. The follower's reaction is quite generally represented as the solution set to a monotone variational inequality. For the solution of this nonconvex mathematical program a penalty approach is proposed, based on the formulation of the lower level variational inequality as a mathematical program. Under natural regularity conditions, we prove the exactness of a certain penalty function, and give strong necessary optimality conditions for a class of generalized bilevel programs. Keywords: Optimization  Variational Inequalities  Bilevel Programming  Exact Penalty  Descent Methods 1 Introduction We consider a twolevel hierarchical system in Euclidian space where the upper level decision maker (hereafter the leader) controls a vector of variables x, and the lower level (hereafter the follower) controls a vector of variables y. The leader makes its decision first...
A Mathematical Model and Descent Algorithm for Bilevel Traffic Management
, 2002
"... We provide a new mathematical model for strategic traffic management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC). The model includes two types of control (upperlevel) variables, which may be used to describe such traffic management actions as traffic signa ..."
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Cited by 7 (4 self)
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We provide a new mathematical model for strategic traffic management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC). The model includes two types of control (upperlevel) variables, which may be used to describe such traffic management actions as traffic signal setting, network design, and congestion pricing. The lowerlevel problem of the MPEC describes a traffic equilibrium model in the sense of Wardrop, in which the control variables enter as parameters in the travel costs. We consider a (small) variety of model settings, including fixed or elastic demands, the possible presence of side constraints in the traffic equilibrium system, and representations of traffic flows and management actions in both linkroute and linknode space. For this model, we also propose and analyze a descent algorithm. The algorithm utilizes a new reformulation of the MPEC into a constrained, locally Lipschitz minimization problem in the product space of controls and traffic flows. The reformulation is based on the Minty (1967) parameterization of the graph of the normal cone operator for the traffic flow polyhedron. Two immediate advantages of making use of this reformulation are that the resulting descent algorithm can be operated and established to be convergent without requiring that the travel cost mapping is monotone, and without having to ever solve the lowerlevel equilibrium problem. We provide example realizations of the algorithm, establish their convergence, and interpret their workings in terms of the traffic network.
Sensitivity analysis of separable traffic equilibrium equilibria, with application to bilevel optimization in network design
, 2005
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Sensitivity Analysis of Variational Inequalities Over Aggregated Polyhedra, With Application to Traffic Equilibria
, 2000
"... Some instances of variational inequality models over polyhedral sets can be stated by either using disaggregated variables or using aggregated variables through an affine transformation. For such problems, we establish that sensitivity analysis results under ample parameterizations rely neither on t ..."
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Cited by 2 (2 self)
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Some instances of variational inequality models over polyhedral sets can be stated by either using disaggregated variables or using aggregated variables through an affine transformation. For such problems, we establish that sensitivity analysis results under ample parameterizations rely neither on the strict monotonicity properties of the problem in terms of the disaggregated variables, nor on any particular choice of their values at the solution. We show how to utilize the affine transformation to devise computational tools for calculating sensitivity results, and apply them to the sensitivity analysis of elastic demand traffic equilibrium problems. Key words: Disaggregated representation; aggregated representation; ample parameterization; sensitivity analysis; locally Lipschitz function; directional derivative; critical cone; Wardrop; traffic equilibrium. 1 Introduction The paper concerns the sensitivity analysis of variational problems of the form \Gammaf (ae; x) 2 NC (x); (1) w...
Entropic Regularization Approach for Mathematical Programs with Equilibrium Constraints
, 2003
"... A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computational study on a set of wellknow ..."
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Cited by 1 (0 self)
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A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computational study on a set of wellknown test problems.
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"... Date prepared 3/16/2012DISCLAIMER AND ACKNOWLEDGMENT The contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein. This document is disseminated under the sponsorship of the Department of Transportation Universit ..."
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Date prepared 3/16/2012DISCLAIMER AND ACKNOWLEDGMENT The contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein. This document is disseminated under the sponsorship of the Department of Transportation University Transportation Centers Program, in the interest of information exchange. The U.S. Government assumes no liability for the contents or use thereof.