Optimization and variational inequalities with pseudoconvex functions
type: Item
instance of: scholarly article
title: Optimization and variational inequalities with pseudoconvex functions
zbMATH Open document ID: 1229.90198
author: V. Ivanov
published in: Journal of Optimization Theory and Applications
publication date: 2010-12-21T00:00:00Z
review text: The author defines a class of pseudoconvex functions in an extended sense and a notion of stationarity in an extended sense, which is called stationarity of order infinity. The inf-stationary points of infinite order are called extended inf-stationary points. Then it is proved that each global minimizer of a function \(f\) over a convex set \(X\) is an extended inf-stationary point. By defining a new variational inequality of Stampacchia type whose solution set is denoted by \(S(f,X)\) it is proved that \(S(f,X)\) gives an approximation of the set of global minimizers \(GM(f,X)\) of \(f\) over \(X\). Then, it is shown that the pseudoconvex functions in the extended sense are the largest class such that \(S(f,Y)\) coincides with \(GM(f,Y)\) for all convex subsets \(Y\) of \(X\). In the last part of the paper the author introduces a notion of pseudomonotone lower Dini directional derivatives in an extended sense using higher-order Dini derivatives and proves that a radially lower semicontinuous function is pseudoconvex in the extended sense if and only if its Dini derivatives are pseudomonotone in the extended sense. It is proved that the radially lower semicontinuous pseudoconvex functions in the extended sense are the largest class such that \(S(f,Y)=M(f,Y)\) for all convex subsets \(Y\) of \(X\), where \(M(f,Y)\) denotes the solution set of the Minty variational inequality of differential type over \(Y\). Finally, the paper deals with the characterizations of the solution set of nonlinear programming problems, whose objective functions are pseudoconvex in the extended sense.
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 5828739
zbMATH Keywords: nonsmooth optimization
zbMATH Keywords: pseudo-convex functions
zbMATH Keywords: variational inequalities of differential type
zbMATH Keywords: characterizations of the solution set of the nonlinear programming problem
reviewed by: Francisco Guerra Vazquez
MaRDI profile type: Publication
full work available at URL: s10957-010-9682-5
OpenAlex ID: W2092972652
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DOI: 10.1007/S10957-010-9682-5
description: scientific article; zbMATH DE number 5828739