Mosco stability of proximal mappings in reflexive Banach spaces

Published: 2006/04/25

Convex Optimization bregman distance, legendre function, modulus of total convexity, mosco convergence of a sequence of functions, proximal mapping relative to a convex function, relative projection onto a convex set, uniformly convex function Short URL: https://optimization-online.org/?p=9912

In this paper we establish criteria for the stability of the proximal mapping \textrm{Prox} $_{\varphi }^{f}=(\partial \varphi +\partial f)^{-1}$ associated to the proper lower semicontinuous convex functions $\varphi $ and $f$ on a reflexive Banach space $X.$ We prove that, under certain conditions, if the convex functions $\varphi _{n}$ converge in the sense of Mosco to $\varphi $ and if $\xi _{n}$ converges to $\xi ,$ then \textrm{Prox} $_{\varphi _{n}}^{f}(\xi _{n})$ converges to \textrm{Prox} $_{\varphi }^{f}(\xi ).$

Mosco stability of proximal mappings in reflexive Banach spaces

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