Nonlinear programming and variational inequality problems. A unified approach
type: Item
instance of: scholarly article
title: Nonlinear programming and variational inequality problems. A unified approach
zbMATH Open document ID: 0913.65058
author: Michael Patriksson
published in: Applied Optimization
publication date: 1999-02-11T00:00:00Z
review text: The book proposes a unified approach of variational inequality problems and of nonlinear programming through the utilization of some algorithms from the same class which allow the author formulates a framework algorithm named ``cost approximation algorithm'' (shortly ``CA algorithm''). Let \(X\subseteq \mathbb{R}^n\) be a nonempty closed convex set, \(u:\mathbb{R}^n\to \mathbb{R}\cup \{+\infty\}\) be a lower semicontinuous proper and convex function and \(F: \text{dom }u\times X\to \mathbb{R}^n\) be a continuous vector-valued mapping. The main problem approached in this book is: ``\([\text{GVIP}(F, u,X)]\): Find a vector \(x^*\in \mathbb{R}^n\) such that \(F(x^*)+\partial u(x^*)+ N_X(x^*)\ni 0^n\)''. Here \(\partial u(x^*)\) is the Moreau subdifferential of \(u\) at \(x^*\), \(N_X(x^*)\) is the normal cone at \(X\) in \(x^*\) and \(0^n\) is the null element of \(\mathbb{R}^n\). This problem is known as a ``generalized variational inequality'', or as a ``nonlinear variational inequality'' or as a ``generalized equation''. This problem and its various special cases have a large variety of applications in the mathematical and engineering science. The book is divided into nine chapters. 1. Introduction (p. 1-38). Various specially cases of \([\text{GVIP}(F, u,X)]\) problem are presented, as: saddle point problem, nonlinear programming problem, penalty function problem, Nash equilibrium problem in games etc. It is described also the ``CA algorithm''. 2. Technical preliminaries (p. 39-56). The cases when the problem \([\text{GVIP}(F,u,X)]\) admits solutions are presented. The CA algorithm is used in the resolution of a variational subproblem, and lower and upper bounds of the arising error are estimated. 3. Instances of the cost approximation algorithm (p. 57-94). Some algorithms of CA type are presented, as: interior point algorithm, Jacobi and Gauss-Seidel methods, regularization and proximal point methods, decomposition-coordination methods, decomposition of optimizati…
reviewed by: Stefan Mititelu
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 49M37
zbMATH DE Number: 1250530
zbMATH Keywords: textbook
zbMATH Keywords: generalized variational inequality
zbMATH Keywords: nonlinear variational inequality
zbMATH Keywords: saddle point problem
zbMATH Keywords: penalty function problem
zbMATH Keywords: Nash equilibrium problem
zbMATH Keywords: interior point algorithm
zbMATH Keywords: Jacobi and Gauss-Seidel methods
zbMATH Keywords: regularization
zbMATH Keywords: proximal point methods
zbMATH Keywords: decomposition-coordination methods
zbMATH Keywords: merit functions
zbMATH Keywords: convergence
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1250530