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 one-level 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.

. 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, first-level constraints are linear, and second-level (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, ill-conditioning, convexity of the objective, monotonicity and symmetry of the second-level 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...

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 two-level 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...

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 (upper-level) variables, which may be used to describe such traffic management actions as traffic signal setting, network design, and congestion pricing. The lower-level 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 link-route and link-node 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 lower-level equilibrium problem. We provide example realizations of the algorithm, establish their convergence, and interpret their workings in terms of the traffic network.

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...

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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 well-known test problems.

Number of pages 17

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.