A CONVEX REPRESENTATION OF MAXIMAL MONOTONE OPERATORS

Abstract

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∗. Throughout this note, X will stand for a Banach space, X for its continuous dual and 〈·, ·〉 for the duality pairing between X and X and defined by 〈x, x〉 = x(x). By the symbol T : X −−→ −→ X ∗ we denote a set-valued mapping T that associates with each x ∈ X a (possibly empty) set T (x) in X. Recall that T is declared monotone if x1 ∈ T (x1), x ∗ 2 ∈ T (x2) =⇒ 〈x1 − x2, x ∗ 1 − x ∗ 2〉 ≥ 0. A monotone mapping T which cannot be properly extended to another monotone mapping is called maximal monotone. Such mappings are very important in variational analysis and optimization. Equivalently, denoting by gph T := {(x, x) ∈ X ×X | x ∈ T (x)} the graph of the set-valued mapping T , maximal monotone operators are those monotone operators satisfying the property : (1) If 〈x− y, x − y〉 ≥ 0 for all (x, x) ∈ gph T, then (y, y) ∈ gph T. A typical example of maximal monotone operator is the subdifferential of a lower semicontinuous convex function; see Rockafellar [1] and the monograph by Simons [3]. Following the standard notations in Convex Analysis (for instance [2]), Γ(X×X) is the set of lower semicontinuous convex functions φ : X×X → R = [−∞,+∞], which are proper, i.e., φ is not identically equal to +∞. Given φ ∈ Γ(X ×X), define b(φ) = {(x, x) ∈ X ×X| φ(x, x) ≤ 〈x, x〉} 1991 Mathematics Subject Classification. 47H05, 49J53 .