Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results (Nonconvex Optimization and Its Applications, 28)

Outrata, Jiri; Kocvara, M.; Zowe, J.

ISBN 10: 0792351703 / ISBN 13: 9780792351702

Published by Springer, 1998

About this Item

Seller Inventory # ABLIING23Feb2416190182927

Bibliographic Details

• Title: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results (Nonconvex Optimization and Its Applications, 28)
• Publisher: Springer
• Publication Date: 1998
• Language: English
• ISBN 10: 0792351703
• ISBN 13: 9780792351702
• Binding: Hardcover
• Condition: New

About this title

Synopsis

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.

Synopsis

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 will also find the book useful.