Necessary optimality conditions for nonsmooth problems with operator constraints
type: Item
instance of: scholarly article
title: Necessary optimality conditions for nonsmooth problems with operator constraints
zbMATH Open document ID: 0662.90088
DOI: 10.4171/ZAA/317
author: H.-P. Scheffler
author: Winfried Schirotzek
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1988-01-01T00:00:00Z
review text: The authors extend Neustadt's theory of upper convex approximations (UCA's) to operators between normed linear spaces; ``upper convexity'' is defined in terms of the ordering induced by a closed convex cone in the range space. A scalarized version of this concept, called the weak UCA, is also introduced, and used to define a set of generalized subdifferentials. Necessary conditions for optimality in an abstract mathematical program are then given, both in terms of UCA's and in terms of the corresponding generalized subdifferentials. These problems are distinguished by their infinite-dimensional inequality constraints - G(x)\(\in L\), where G:E\(\to F\) is a nonlinear operator between normed linear spaces E and F, and L is the closed convex cone of ``nonnegative elements'' of F. The authors give necessary conditions in Fritz John form, and then describe constraint qualifications under which they hold in ``normal'' (KKT) form.
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4083398
zbMATH Keywords: Karush-Kuhn-Tucker conditions
zbMATH Keywords: upper convex approximations
zbMATH Keywords: normed linear spaces
zbMATH Keywords: generalized subdifferentials
zbMATH Keywords: Necessary conditions for optimality
zbMATH Keywords: infinite-dimensional inequality constraints
zbMATH Keywords: nonlinear operator
zbMATH Keywords: closed convex cone
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4083398