The convex function determined by a multifunction (original) (raw)

Maximal monotone operators, convex functions and a special family of enlargements

2002

This work establishes new connections between maximal monotone operators and convex functions. Associated to each maximal monotone operator, there is a family of convex functions, each of which characterizes the operator. The basic tool in our analysis is a family of enlargements, recently introduced by Svaiter. This family of convex functions is in a one-to-one relation with a subfamily of these enlargements. We study the family of convex functions, and determine its extremal elements.

A Convex Representation of Maximal Monotone Operators

Journal of Nonlinear …, 2001

In this note, we prove that the set of maximal monotone operators between a normed linear space X and its continuous dual X * can be identified as some subset of the set Γ(X × X * ) of all lower semicontinuous convex proper functions on X × X * .

Operating enlargements of monotone operators: new connections with convex functions

Pacific Journal of Mathematics

Given a maximal monotone operator T in a Banach space, a family of enlargements E(T) of T has been introduced by Svaiter. He also defined a sum and a positive scalar multiplication of enlargements. The first aim of this work is to further study the properties of these operations. Burachik and Svaiter studied a family of convex functions H(T) which is in a one to one correspondence with E(T). The second

Variational Principles for Monotone and Maximal Bifunctions

2003

We establish variational principles for monotone and maximal bifunctions of Brøndsted-Rockafellar type by using our characterization of bifunction's maximality in reflexive Banach spaces. As applications, we give an existence result of saddle point for convex-concave function and solve an approximate inclusion governed by a maximal monotone operator.

Regular Maximal Monotone Operators

1998

The purpose of this paper is to introduce a class of maximal monotone operators on Banach spaces that contains all maximal monotone operators on reflexive spaces, all subdifferential operators of proper, lsc, convex functions, and, more generally, all maximal monotone operators that verify the simplest possible sum theorem. Dually strongly maximal monotone operators are also contained in this class. We shall prove that if T is an operator in this class, then dom(T) (the norm closure of its domain) is convex, the interior of co(dom(T)) (the convex hull of the domain of T) is exactly the set of all points of dom(T) at which T is locally bounded, and T is maximal monotone locally, as well as other results.

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

2010

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T, S which share the same convex-like domain D coincide whenever T(x) and S(x) have a nonempty intersection for every x in D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their epsilon-subdifferentials intersect at every point of that domain.

Second Order Cones for Maximal Monotone Operators via Representative Functions

Set-Valued Analysis, 2008

It is shown that various …rst and second order derivatives of the Fitzpatrick and Penot repre- sentative functions for a maximal monotone operator T, in a re‡exive Banach space, can be used to represent dierential information associated with the tangent and normal cones to the GraphT. In particular we obtain formula for the Proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of Proto-dierentiability to the graph of T, is often associated with single valuedness of T.