Intersection theorems with applications in optimization
type: Item
instance of: scholarly article
title: Intersection theorems with applications in optimization
zbMATH Open document ID: 1440.47045
author: Ravi P. Agarwal
author: Mircea Balaj
author: Donal O'Regan
published in: Journal of Optimization Theory and Applications
publication date: 2018-11-27T00:00:00Z
review text: The classical Fan-KKM lemma gives a nonempty intersection result expressed by a KKM-mapping \(S\): \(\bigcap_{x\in X} \not= \emptyset\). Originally, a KKM-mapping is a set-valued function \(S: X \twoheadrightarrow Y\), \(X\), \(Y\) nonempty subsets of a vector space, \(X\) convex, if for every nonempty finite subset \(A\) of \(X\) \[ \operatorname{conv}(A) \subset \bigcup_{x \in A}S(x).\tag{1} \] The point of the paper is, that the authors use an interesting generalization of that KKM-mapping (following [\textit{G. Botelho} et al., Rev. Mat. Complut. 23, No. 1, 139--161 (2010; Zbl 1210.46031)] and especially [\textit{G. Botelho} et al., Rev. Mat. Complut. 23, No. 1, 139--161 (2010; Zbl 1210.46031)]) and prove two new intersection theorems. Then they apply them for problems in optimization and variational analysis. The mentioned generalization is, that a second set-valued mapping \(T:X \twoheadrightarrow Y\) is used such that (1) is replaced by \[ T(\operatorname{conv}(A)) \subset \bigcup_{x \in A}S(x),\tag{2} \] or replaced by: for each subset \(A\) as above and any \(x \in \operatorname{conv}(A)\), \[ T(x) \cap S(A) \not= \emptyset. \tag{3} \] (1) is called KKM-mapping w.r.t. \(T\), (2) is called weak KKM-mapping w.r.t. \(T\). Both new intersection theorems give the nonempty intersection that there is \(x_0\in X\) such that \(T(x_0 \cap \bigcap_{u\in X}S(u) \not= \emptyset\) under the conditions that \(X, Y\) are sets in topological vector spaces and, in the first theorem, \(X\) nonempty, convex, compact, \(Y\) nonempty, convex, \(S\) a closed set-valued mapping with convex values and convex cofibers and weak-KKM w.r.t. \(T\), \(T\) has compact convex values and, finally, for each \(u\in X\), the set \(\{x\in X: T(x)\cap S(u)\not=\emptyset\}\) is closed. The second intersection theorem is a noncompact version of the first one and also with weak-KKM \(S\). The content-rich applications show that the skillful use of \(T\) in addition to \(S\) leads to improved soluti…
reviewed by: Alfred Göpfert
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 47N10
zbMATH DE Number: 6985493
zbMATH Keywords: weak KKM map
zbMATH Keywords: variational inequality
zbMATH Keywords: minimax inequality
zbMATH Keywords: intersection theorem
zbMATH Keywords: complementarity problem
MaRDI profile type: Publication
full work available at URL: s10957-018-1331-4
OpenAlex ID: W2808134898
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DOI: 10.1007/S10957-018-1331-4
description: scientific article; zbMATH DE number 6985493