Representative functions of maximally monotone operators and bifunctions

• 成果类型: Article
• 署名作者: Bianchi, Monica; Hadjisavvas, Nicolas; Pini, Rita
• 署名单位: Catholic University of the Sacred Heart; King Fahd University of Petroleum & Minerals; University of Milano-Bicocca
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1020-8
• 发表日期: 2018
• 页码: 433-448
• 关键词: convex-functions; fitzpatrick functions

摘要

The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has as Fitzpatrick transform. In the special case where is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In this way we exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.