Publisher Description

This book presents an in-depth study and a solution technique for an important class of optimization problems. This class is characterized by special constraints - parameter-dependent convex programs, variational inequalities, or complementarity problems. All these so-called equilibrium constraints are mostly treated in a convenient form of generalized equations. The book begins with a chapter on auxiliary results followed by a description of the main numerical tools - a bundle method of nonsmooth optimization and a nonsmooth variant of Newton's method. Following this, stability and sensitivity theory for generalized equations is presented, based on the concept of strong regularity. This enables one to apply the generalized differential calculus for Lipschitz maps to derive optimality conditions and to arrive at a solution method. A large part of the book focuses on applications coming from continuum mechanics and mathematical economy. A series of nonacademic problems is introduced and analyzed in detail. Each problem is accompanied with examples that show the efficiency of the solution method.This book is addressed to applied mathematicians and engineers working in continuum mechanics, operations research and economic modelling. Students interested in optimization should also find the book useful.

Table of Contents

I Theory.- 1. Introduction.- 2. Auxiliary Results.- 3. Algorithms of Nonsmooth Optimization.- 4. Generalized Equations.- 5. Stability of Solutions to Perturbed Generalized Equations.- 6. Derivatives of Solutions to Perturbed Generalized Equations.- 7. Optimality Conditions and a Solution Method.- II Applications.- 8. Introduction.- 9. Membrane with Obstacle.- 10. Elasticity Problems with Internal Obstacles.- 11. Contact Problem with Coulomb Friction.- 12. Economic Applications.- Appendices.- A-Cookbook.- A.1 Problem.- A.2 Assumptions.- A.3 Formulas.- B-Basic facts on elliptic boundary value problems.- B.1 Distributions.- B.2 Sobolev spaces.- B.3 Elliptic problems.- C-Complementarity problems.- C.1 Proof of Theorem 4.7.- C.2 Supplement to proof of Theorem 4.9.- References.

Long Description

In the early fifties, applied mathematicians, engineers and economists started to pay c10se attention to the optimization problems in which another (lower-Ievel) optimization problem arises as a side constraint. One of the motivating factors was the concept of the Stackelberg solution in game theory, together with its economic applications. Other problems have been encountered in the seventies in natural sciences and engineering. Many of them are of practical importance and have been extensively studied, mainly from the theoretical point of view. Later, applications to mechanics and network design have lead to an extension of the problem formulation: Constraints in form of variation al inequalities and complementarity problems were also admitted. The term "generalized bi level programming problems" was used at first but later, probably in Harker and Pang, 1988, a different terminology was introduced: Mathematical programs with equilibrium constraints, or simply, MPECs. In this book we adhere to MPEC terminology. A large number of papers deals with MPECs but, to our knowledge, there is only one monograph (Luo et al. , 1997). This monograph concentrates on optimality conditions and numerical methods. Our book is oriented similarly, but we focus on those MPECs which can be treated by the implicit programming approach: the equilibrium constraint locally defines a certain implicit function and allows to convert the problem into a mathematical program with a nonsmooth objective.

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