An application of the bivariate inf-convolution formula to enlargements of monotone operators
type: Item
instance of: scholarly article
title: An application of the bivariate inf-convolution formula to enlargements of monotone operators
zbMATH Open document ID: 1188.47040
author: Radu I. Boţ
author: Ernö Robert Csetnek
published in: Set-Valued Analysis
publication date: 2009-03-16T00:00:00Z
review text: This paper extends to the framework of enlargements a property concerning the \(\varepsilon\)-subdifferential of the sum of two proper, convex and lower semicontinuous functions \(f, g: X \to\mathbb R\cup\{+\infty\}\) such that \(\operatorname{dom}f\cap \operatorname{dom} g\not= \emptyset\). This property, established in [\textit{R.\,S.\thinspace Burachik} and \textit{V.\,Jeyakumar}, ``A new geometric condition for Fenchel's duality in infinite dimensional spaces'', Math.\ Program.\ 104, No.\,2--3 (B), 229--233 (2005; Zbl 1093.90077)], states the equivalence between the following statements. (i) \(\operatorname{epi}f^*+ \operatorname{epi}g^*\) is closed in \((X^*, \omega(X^*, X)) \times\mathbb R\). (ii) \(\partial_{\varepsilon}( f + g)(x) = \bigcup_{\varepsilon_1,\varepsilon_2\geq 0,\, \varepsilon_1+\varepsilon_2=\varepsilon} (\partial_{\varepsilon_1} f(x)+ \partial_{\varepsilon_2} g(x))\) for all \(\varepsilon\geq 0\) and for all \(x\in X\). This equivalence was first proved in [loc.\ cit.] in the framework of Banach spaces, and extended to real separated locally convex spaces in [\textit{R.\,I.\thinspace Boţ,} \textit{I.\,B.\thinspace Hodrea} and \textit{G.\,Wanka}, ``\(\varepsilon\)-optimality conditions for composed convex optimization problems'', J.~Approx.\ Theory 153, No.\,1, 108--121 (2008; Zbl 1158.46029)]. The authors give a necessary and sufficient condition that guarantees, in the case of enlargements of maximal monotone operators, an equivalence analogous to the one stated above. This is achieved by giving a necessary and sufficient condition for a bivariate inf-convolution formula \[ (h_1{\square}_2 h_2)^* = {h_1}^*{\square}_1 {h_2}^*, \] where \(h_1, h_2 : X \times Y \to\mathbb R\cup\{+\infty\}\) are proper, convex and lower semicontinuous functions defined in the product of two real separated locally convex spaces. Using the above formula for convex functions representing maximal monotone operators, the authors establish conditions ensuring the maxim…
reviewed by: R. S. Burachik
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 42A50
zbMATH DE Number: 5529279
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: enlargements of maximal monotone operators
zbMATH Keywords: Fitzpatrick function
zbMATH Keywords: convex function associated with a maximal monotone operator
zbMATH Keywords: bivariate inf-convolution
MaRDI profile type: Publication
full work available at URL: s11228-008-0093-9
OpenAlex ID: W2042277485
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DOI: 10.1007/S11228-008-0093-9
description: scientific article; zbMATH DE number 5529279