Remarks on infinite dimensional duality
type: Item
instance of: scholarly article
title: Remarks on infinite dimensional duality
zbMATH Open document ID: 1198.90390
published in: Journal of Global Optimization
publication date: 2010-04-28T00:00:00Z
review text: The convex like optimization problem \(f(x)\rightarrow \min , g(x)\in -C\) with \(x\) in a TVS and \(g(x)\) in a normed space, \(C\) a convex order cone is considered. A strong (Lagrange) duality result derived in [\textit{P. Daniele, S. Giuffré, G. Idone} and \textit{A. Maugeri}, Math. Ann. 339, No. 1, 221--239 (2007; Zbl 1356.49064)] uses the empty intersection of the tangent cone at the optimal point w.r.t. \(\tilde M\) with \((0,\infty)\times \{0\}\) as constrained qualification. Hereby is \(\tilde M\) the Minkowski sum of the used order cone \(\mathbb{R}_+ \times C\) and the image of the non feasible points by \((f,g)\). Let now \(M\supset \{(0,0)\}\) be the Minkowski sum of the order cone \(\mathbb{R}_+ \times C\) and the whole image of \((f-f(x_0),g)\). In the paper is shown, that the same duality result holds when the above constrained qualification is replaced by the following condition: There is an element \(\xi\) in \(M\) and an outer normal \(\eta^*\) from the dual space at zero of \(M\) such that \(\eta^* \xi <0\). Hence ``the interior point free'' primal constrained qualification can be replaced by some strong inequality with respect to the outer normal cone. The given approach works for spaces where the order cone of the constraint \(g\) does not contain interior points e.g. for \(l^2\) with the order cone of nonnegative sequences.
reviewed by: Armin Hoffmann
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 49N15
zbMATH DE Number: 5702524
zbMATH Keywords: strong duality
zbMATH Keywords: quasi-relative interior
zbMATH Keywords: tangent cone
zbMATH Keywords: normal cone
author: Antonino Maugeri
author: Fabio Raciti
MaRDI profile type: Publication
full work available at URL: s10898-009-9442-y
OpenAlex ID: W2091710051
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DOI: 10.1007/S10898-009-9442-Y
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description: scientific article; zbMATH DE number 5702524