Nonsmooth approach to optimization problems with equilibrium constraints. Theory, applications and numerical results
type: Item
instance of: scholarly article
title: Nonsmooth approach to optimization problems with equilibrium constraints. Theory, applications and numerical results
zbMATH Open document ID: 0947.90093
author: Michal Kočvara
author: J. Zowe
author: Jiří Outrata
published in: Nonconvex Optimization and Its Applications
publication date: 1998-09-07T00:00:00Z
review text: The present monogragh grew out of a series of papers on mathematical programs with equilibrium constraints published by the authors in the years 1993-1997. It is focused on the following optimization problem: minimize \(f(x,z)\) subject to \(z\in S(x)\) nad \(x\in U_{ad}\). Here \(f\) maps \(\mathbb{R}^n \times\mathbb{R}^k\) into \(\mathbb{R}\), \(U_{ad}\) is a nonempty closed subset of \(\mathbb{R}^n\), and \(S\) is the solution map of the perturbed generalized equation \[ 0\in C(x,z)+ N_Q(z), \] where \(C:\mathbb{R}^n\times \mathbb{R}^k \to\mathbb{R}^k\) is a continuous map, \(Q\) is a nonempty closed convex subset of \(\mathbb{R}^k\), and \(N_Q (z)\) is the normal cone to \(Q\) at \(z\in\mathbb{R}^k\). The relation \(z\in S(x)\) is called equilibrium constraint. Supposing that this constraint locally defines a certain implicit function, the authors convert the considered problem into an ``easier'' mathematical programming problem with a nonsmooth objective function. By applying the ``non-differentiable'' calculus of Clarke to the new problem, they derive necessary optimality conditions for the initial problem. Furthermore, by using the bundle method of nonsmooth optimization for the new problem, they get a solution technique for the initial problem. The efficiency of this approach is shown on a series of tough nonacademic problems from the area of optimum shape design and economic modelling. Dealing with both, the theory and applications of optimization problems with equilibrium constraints, this book is extremely useful for researchers in optimization theory and for applied mathematicians.
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 90-02
zbMATH DE Number: 1197245
zbMATH Keywords: equilibrium constraints
zbMATH Keywords: nonsmooth objective function
zbMATH Keywords: bundle method of nonsmooth optimization
zbMATH Keywords: optimum shape design
zbMATH Keywords: economic modelling
reviewed by: Wolfgang W. Breckner
describes a project that uses: SNOPT
describes a project that uses: LANCELOT
describes a project that uses: NLPQL
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1197245