Maximal monotone operators, convex functions and a special family of enlargements (original) (raw)

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 * .

Monotone Operators Without Enlargements

Springer Proceedings in Mathematics & Statistics, 2013

Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the sum of two maximally monotone linear relations. We also present a new proof of the maximality of the sum of a maximally monotone linear relation and a normal cone operator when the domain of the linear relation intersects the interior of the domain of the normal cone.

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.

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 the Maximal Extensions of Monotone Operators and Criteria for Maximality

Within a nonzero, real Banach space we study the problem of characterising a maximal extension of a monotone operator in terms of minimality properties of representative functions that are bounded by the Penot and Fitzpatrick functions. We single out a property of this space of representative functions that enable a very compact treatment of maximality and pre-maximality issues.

Faces and Support Functions for the Values of Maximal Monotone Operators

Journal of Optimization Theory and Applications, 2020

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on uniformly Banach spaces with uniformly convex duals, or their domains have nonempty interiors on reflexive real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains.

ε-enlargements of maximal monotone operators in Banach spaces

1999

Given a maximal monotone operator T in a Banach space, we consider an enlargement T ε , in which monotonicity is lost up to ε, in a very similar way to the ε-subdifferential of a convex function. We establish in this general framework some theoretical properties of T ε , like a transportation formula, local Lipschitz continuity, local boundedness, and a Brøndsted & Rockafellar property.

Enlargements and sums of monotone operators

Nonlinear Analysis: Theory, Methods & Applications, 2002

In this paper we study two important notions related to monotone operators. One is the concept of enlargement of a given monotone operator which has turned out to be a useful tool in the analysis of approximate solutions to problems involving monotone operators. The second one is the notion of sum of monotone operators. For the latter we introduce and study a kind of extended sum of two monotone operators, which, in several important cases, turns out to be a maximal monotone operator.