# Representative functions of maximally monotone operators and bifunctions

Representative functions of maximally monotone operators and bifunctions 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 $$\varphi$$ φ 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 $$\varphi$$ φ as Fitzpatrick transform. In the special case where $$\varphi$$ φ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

# Representative functions of maximally monotone operators and bifunctions

, Volume 168 (2) – May 11, 2016
16 pages

/lp/springer_journal/representative-functions-of-maximally-monotone-operators-and-9VaqEsx0iu
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-016-1020-8
Publisher site
See Article on Publisher Site

### Abstract

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 $$\varphi$$ φ 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 $$\varphi$$ φ as Fitzpatrick transform. In the special case where $$\varphi$$ φ 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.

### Journal

Mathematical ProgrammingSpringer Journals

Published: May 11, 2016

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