Further generalizations of Farkas' theorem and their applications in optimal control
type: Item
instance of: scholarly article
title: Further generalizations of Farkas' theorem and their applications in optimal control
zbMATH Open document ID: 0902.49020
author: V. N. Phat
author: Jong Yeoul Park
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: A set-valued generalization is given for Farkas' theorem in Banach spaces. If \(S\) and \(K\) are convex cones, \(A\) linear, and \(T\) a convex process with \(T(S)\) closed, then \[ [(\forall x\in K)(\exists z\in S) Ax\in T(z)]\Leftrightarrow [T^*(y^*)\cap S^*\neq\emptyset\Rightarrow A^*y^*\in K^*]. \] A generalization is given without the closedness assumption on \((T(S))\). The generalized Gale alternative theorem in nonlinear programming is a corollary. The results are applied to constrained controllability theory in Banach spaces, as well as to some multiobjective optimization problems.
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 90C29
Mathematics Subject Classification ID: 90C05
zbMATH DE Number: 1115520
zbMATH Keywords: set-valued mapping
zbMATH Keywords: Farkas' theorem
zbMATH Keywords: convex process
zbMATH Keywords: Gale alternative theorem
zbMATH Keywords: constrained controllability
zbMATH Keywords: multiobjective optimization
reviewed by: B. D. Craven
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5612
OpenAlex ID: W2020594720
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DOI: 10.1006/JMAA.1997.5612
description: scientific article; zbMATH DE number 1115520