An application of the bivariate inf-convolution formula to enlargements of monotone operators
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 maximality of the sum of two maximal monotone operators defined on a reflexive Banach space. This formula is also used by the authors in Theorem 11 to give a so-called closedness-type regularity condition which completely characterizes the enlargement of the sum of two maximal monotone operators in terms of the enlargements of the maximal monotone operators involved.
enlargements of maximal monotone operators
convex function associated with a maximal monotone operator
scientific article; zbMATH DE number 5529279
scientific article