Characterizations of convex approximate subdifferential calculus in Banach spaces (Q2790639)

For a proper convex lsc function \(f:X\to\mathbb R\cup\{\infty\}\) defined on a real normed space \(X\) consider: {\parindent=6mm \begin{itemize} \item[{\(\bullet\)}] the Moreau-Yosida envelope:\; \( f_\lambda(x):=\inf_{y\in X}\left[f(y)+(2\lambda)^{-1}\|x-y\|^2\right],\;\; x\in X\,,\lambda >0;\) \item [{\(\bullet\)}] the \(\varepsilon\)-subdifferential:\; \(\partial_\varepsilon f(x):=\{x^*\in X^* : \langle x^*,y-x\rangle\leq f(x)-f(y),\, \forall y\in X\}\,,\) and \item [{\(\bullet\)}] the Fenchel conjugate:\; \(f^*:X^*\to\mathbb R\cup\{\infty\}\), \( f^*(x^*):=\sup_{y\in X}\left[\langle x^*,y\rangle-f(y)\right],\;\; x^*\in X^*\,.\) NEWLINENEWLINE\end{itemize}} It follows that \(\partial_0f(x)=\partial f(x)\) (the usual subdifferential) and NEWLINE\[NEWLINE\partial_\varepsilon f(x)=\{x^*\in X^*:f(x)+f^*(x^*)\leq \langle x^*,x\rangle+\varepsilon\}.NEWLINE\]NEWLINE The inf-convolution of two proper convex lsc functions \(f,g:X\to\mathbb R\cup\{\infty\}\) is defined by NEWLINE\[NEWLINE(f\square g)(x):=\inf_{y\in X}\left[f(y)+g(x-y)\right],\, x\in X.NEWLINE\]NEWLINE It follows that \(\displaystyle (f\square g)^*=f^*+g^*\) and \(\displaystyle \left((2\lambda)^{-1}\|\cdot\|^2\right)^*(x^*)=(\lambda/2)\|x^*\|\), so that NEWLINE\[NEWLINE\displaystyle f_\lambda = f\square (2\lambda)^{-1}\|\cdot\|^2 NEWLINE\]NEWLINE and NEWLINE\[NEWLINE f^*_\lambda(x^*)=f^*(x^*)+(\lambda/2)\|x^*\|^2\,.NEWLINE\]NEWLINENEWLINENEWLINEThe aim of the present paper is to emphasize ``the natural and useful role of the Moreau-Yosida envelopes within the theory of subdifferential calculus of convex functions. This will allow simple and nice proofs of well-known formulas.''NEWLINENEWLINEThe paper relies on some formulas relating these notions contained in three preliminary theorems labeled A,B,C. As a sample we quote the formula (the simplest) from Theorem A NEWLINE\[NEWLINE \partial_\varepsilon f(x) = \bigcap_{\delta>\varepsilon}\bigcup_{\eta>0}\bigcap_{0<\lambda<\eta}\partial_\delta f_\lambda(x)\,,\;\; x\in X,\;\varepsilon\geq 0\,.NEWLINE\]NEWLINENEWLINENEWLINEBased on these formulas and ``relying on the continuity behavior of the inf-convolution of their corresponding Fenchel conjugates, it will be established that the approximate subdifferential of \(f + g\) can be written as the \(\tau\)-limit of the sum of the approximate subdifferentials of some appropriate functions \(f_k\) and \(g_k\) at the reference point, where \(\tau\) is any given topology intermediate between the weak\(^*\) and the norm topologies on the dual space \dots Such a condition turns out to also be necessary in Banach spaces. These results extend both the classical formulas by Hiriart-Urruty and Phelps and by Thibault.''