Multivalued analysis and nonlinear programming problems with perturbations
type: Item
instance of: scholarly article
title: Multivalued analysis and nonlinear programming problems with perturbations
zbMATH Open document ID: 1061.90105
author: B. Luderer
author: Tatyana Satsura
author: Leonid Minchenko
published in: Nonconvex Optimization and Its Applications
publication date: 2003-01-20T00:00:00Z
review text: The aim of this book is to study, in finite dimensional spaces, multivalued mappings and the associated marginal functions and to apply the results to optimization problems with quite general constraints. More precisely, the following class of programming problems is the final object of investigation: \[ (\overline P_x)\qquad f(x,y)\to\inf_y,\;y\in F(x). \] Here \(f:\mathbb{R}^n\times \mathbb{R}^m\to\mathbb{R}\) and \(F\) is a multivalued mapping of \(\mathbb{R}^n\) to \(\mathbb{R}^m\). To a great extent, the analysis of such problems is sensitivity analysis, i.e., it consists in studying the marginal function (or optimal value function) \[ \varphi (x)=\inf\bigl\{f(x,y),\mid y\in F(x)\bigr\} \] and the set of optimal solutions \[ \omega(x)=\bigl\{y\in F(x)\mid\varphi(x)=f(x,y)\bigr\}. \] These functions are in general not differentiable, and this is the starting point of the book. The text is divided into six chapters. After a list with the notation used, Chapter 2 contains elements of convex analysis and of quasidifferentiable calculus in the sense of Demyanov and Rubinov. Chapter 3 deals with continuity and differentiability properties of multivalued mappings. Several notions of semicontinuity and (pseudo-)Lipschitz continuity are introduced and their relationship to each other and to the associated marginal function is studied. Moreover, various kinds of directional derivatives of multivalued mapppings are defined via tangent cones and are investigated in terms of the distance function. Chapter 4 is devoted to subdifferentials of marginal functions. In particular, estimates (in the sense of set inclusion) are established for the Clarke subdifferential and for subdifferentials of Mordukhovich. Chapter 5 presents sufficient conditions for the existence of the directional derivative of marginal functions. In this connection, weakly uniformly differentiable as well as strongly differentiable multifunctions are considered. The concluding chapter deals with the sensiti…
reviewed by: Winfried Schirotzek
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 49-02
zbMATH DE Number: 1855753
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1855753