Subdifferential representation of convex functions on \(X^\)

In this paper, we obtain subdifferential representation of a proper \(w^*\)-lower semicontinous convex function on \(X^*\) as follows: Let \(g\) be a proper convex \(w^*\)-lower semicontinuous function on \(X^*\). Assume that int dom \(g\) \(\neq\emptyset\) (resp. int (dom (\(g^*|_X)\))\(\neq\emptys...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-11
Main Author: Dai, Duanxu
Format: Article
Language:English
Subjects:
Integers
Radon
Representations
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this paper, we obtain subdifferential representation of a proper \(w^*\)-lower semicontinous convex function on \(X^*\) as follows: Let \(g\) be a proper convex \(w^*\)-lower semicontinuous function on \(X^*\). Assume that int dom \(g\) \(\neq\emptyset\) (resp. int (dom (\(g^*|_X)\))\(\neq\emptyset\)). Then given any point \(x_0^*\) \(\in\) D (\(\partial g\cap X\)) and \(x^*\) \(\in\) dom \(g\) (resp. \(x^*\in X^*\)), we have $$g(x^*)=g(x_0^*)+\sup\{\sum_{i=0}^{n-1}\langle x_i,x_{i+1}^*-x_i^*\rangle +\langle x_n,x^*-x_n^*\rangle \},$$ where the above supremum is taken over all integers \(n\), all \(x_i^*\in X^*\) and all \(x_i\in\partial g(x_i^*)\cap X\) for \(i=0,1,\cdots,n\). (resp. if, moreover, \(X^*\) has the Radon-Nikodym property, then we may estimate the above supremum among the set of \(w^*\)-strongly exposed points of \(g\).)
ISSN:2331-8422