[Paper Review] Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces

M. Marques Alves, B. F. Svaiter|ArXiv.org|Sep 23, 2008

Optimization and Variational Analysis10 references17 citations

TL;DR

This paper extends Fitzpatrick's representation theory of maximal monotone operators to non-reflexive Banach spaces by identifying conditions under which a convex function bounded below by the duality product represents a maximal monotone operator. It establishes that if a convex, lower semicontinuous function h satisfies h(x,x*) ≥ ⟨x,x*⟩ and h*(x*,x) ≥ ⟨x,x*⟩ for all (x,x*) ∈ X×X*, then the set T = {(x,x*) | h(x,x*) = ⟨x,x*⟩} is maximal monotone, generalizing earlier results from reflexive to non-reflexive settings under a domain regularity condition.

ABSTRACT

Maximal monotone operators on a Banach space into its dual can be represented by convex functions bounded below by the duality product. It is natural to ask under which conditions a convex function represents a maximal monotone operator. A satisfactory answer, in the context of reflexive Banach spaces, has been obtained some years ago. Recently, a partial result on non-reflexive Banach spaces was obtained. In this work we study some others conditions which guarantee that a convex function represents a maximal monotone operator in non-reflexive Banach spaces.

Motivation & Objective

• To generalize the characterization of maximal monotone operators via convex functions from reflexive to non-reflexive Banach spaces.
• To identify necessary and sufficient conditions for a convex function on X×X* to represent a maximal monotone operator in non-reflexive settings.
• To address the open question of whether minimal functions in the Fitzpatrick family correspond to maximal monotone operators in non-reflexive spaces.
• To establish a regularity condition on the domain of h that ensures the representation of maximal monotonicity via level sets of h.

Proposed method

• Uses the Fitzpatrick function φ_T and its conjugate to characterize maximal monotone operators via convex analysis.
• Applies the Fenchel-Legendre conjugate and the duality product π(x,x*) = ⟨x,x*⟩ to define the class of functions bounded below by π.
• Introduces the transformation J: h ↦ h*, which maps the Fitzpatrick family F_T into itself, preserving the representation property.
• Imposes a domain regularity condition: the union over λ>0 of λ·Pr_X(D(h)) is a closed subspace of X, to ensure closedness of the sublevel set.
• Employs the Fenchel-Young inequality and infimal convolution techniques to derive duality relations between h and its conjugate.
• Uses the closedness of the epigraph of h and the weak*-closedness of the sublevel set to establish maximality of the induced operator.

Experimental results

Research questions

• RQ1 Under what conditions on a convex function h on X×X* does the set T = {(x,x*) | h(x,x*) = ⟨x,x*⟩} define a maximal monotone operator in a non-reflexive Banach space?
• RQ2 Can the characterization of maximal monotonicity via the Fitzpatrick function be extended to non-reflexive Banach spaces without requiring the stronger condition h*(x*,x**) ≥ ⟨x*,x**⟩ for x** ∈ X**?
• RQ3 Is every minimal element g of the family of convex functions bounded below by the duality product necessarily the Fitzpatrick function of some maximal monotone operator in non-reflexive spaces?
• RQ4 What regularity conditions on the domain of h ensure that the sublevel set {(x,x*) | h(x,x*) = ⟨x,x*⟩} is maximal monotone?

Key findings

• A convex, lower semicontinuous function h on X×X* satisfying h(x,x*) ≥ ⟨x,x*⟩ and h*(x*,x) ≥ ⟨x,x*⟩ for all (x,x*) ∈ X×X* generates a maximal monotone operator T defined by T = {(x,x*) | h(x,x*) = ⟨x,x*⟩} in non-reflexive Banach spaces.
• The Fitzpatrick function φ_T is the smallest function in the family F_T, and the transformation J preserves membership in F_T, ensuring h ∈ F_T if h is l.s.c.
• The paper proves that if g is a minimal element of the family of convex functions bounded below by the duality product and the union over λ>0 of λ·Pr_X(D(g)) is a closed subspace of X, then g = φ_T for some maximal monotone operator T.
• The domain regularity condition ensures that the sublevel set of h is closed in the strong topology, which is essential for maximality in non-reflexive spaces.
• The paper resolves a partial case of an open question by Martínez-Legaz and Svaiter, showing that under domain regularity, minimal functions in the duality-bounded convex family are Fitzpatrick functions.
• The result generalizes Theorem 1.2 (Burachik and Svaiter) from reflexive to non-reflexive Banach spaces, replacing the stronger condition h*(x*,x**) ≥ ⟨x*,x**⟩ with the weaker h*(x*,x) ≥ ⟨x,x*⟩ and a domain regularity assumption.