Duality gap of the conic convex constrained optimization problems in normed spaces
type: Item
instance of: scholarly article
title: Duality gap of the conic convex constrained optimization problems in normed spaces
zbMATH Open document ID: 1172.90012
author: Liqun Ban
author: Wen Song
published in: Mathematical Programming. Series A. Series B
publication date: 2009-04-24T00:00:00Z
review text: In this interesting paper the authors extend the results of \textit{T. Champion} [Duality gap in convex programming, Math. Program. 99, No.~3 (A), 487--498 (2004; Zbl 1084.90032)] to the problem \(\min f(x)\) s.t. \(x\in C\), \(h(x)\in-S\), where \(X,Y\) are normed spaces, \(C\subset X\) is a closed convex set, \(S\subset Y\) is a closed convex cone with nonempty interior, \(f:X\to \mathbb{R}\cup\{+\infty\}\) is a proper and lower semicontinuous functional, and \(h:X\to Y\) is an \(S\)-convex mapping. For the study of the preceding problem it is considered the condition \({\mathcal D}(y)\) which means that \(\bigcap_{z>y}\overline{\{h\leq z\}\cap C+\ker x^*}=\overline{\{h\leq y\}\cap C+\ker x^*}\) for every \(x^*\in X^*\), where \(z>y\) \((z\geq y)\) means that \(z-y\in\operatorname{int}S\) \((z-y\in S)\) and \(\{h\leq y\}= \{x\in X\mid h(x)\leq y\}.\) The condition \({\mathcal D}(y)\) is an analog of a condition used by T. Champion.
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 5547235
zbMATH Keywords: zero duality gap
zbMATH Keywords: \(S\)-convex mapping
zbMATH Keywords: \({\mathcal D}(y)\) property
zbMATH Keywords: normed spaces
reviewed by: Constantin Zălinescu
MaRDI profile type: Publication
full work available at URL: s10107-008-0207-z
OpenAlex ID: W1985754192
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DOI: 10.1007/S10107-008-0207-Z
description: scientific article; zbMATH DE number 5547235