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Characterizing Maximal Monotone Operators with Unique Representation

Sotiris Armeniakos, Aris Daniilidis

TL;DR

This work characterizes maximal monotone operators with a unique Fitzpatrick representation. It shows that $3$-monotonicity together with a singleton Fitzpatrick family forces such operators to be subdifferentials, i.e., $A=\\partial f$, and, in spaces with the Radon–Nikodým property, provides a detailed structural form of $f$ as a sum of a support term and indicator of convex sets. It also identifies a broad class of subdifferentials with unique representations via explicit geometric data $(K,C,\\mathcal{V})$ and a compatibility condition, and proves that in finite dimensions these assumptions can be relaxed. The results extend prior work on uniquely representable sublinear and indicator-function cases and motivate a conjecture that uniquely representable maximal monotone operators are either subdifferentials or linear skew-symmetric, thereby clarifying the landscape of operators with a singleton Fitzpatrick family.

Abstract

We study maximal monotone operators $A : X \rightrightarrows X^*$ whose Fitzpatrick family reduces to a singleton; such operators will be called uniquely representable. We show that every such operator is cyclically monotone (hence, $A=\partial f$ for some convex function $f$) if and only if it is 3-monotone. In Radon-Nikodým spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator $A=\partial f$ is uniquely representable if and only if $f$ is the sum of a support and an indicator function of suitable convex sets.

Characterizing Maximal Monotone Operators with Unique Representation

TL;DR

This work characterizes maximal monotone operators with a unique Fitzpatrick representation. It shows that -monotonicity together with a singleton Fitzpatrick family forces such operators to be subdifferentials, i.e., , and, in spaces with the Radon–Nikodým property, provides a detailed structural form of as a sum of a support term and indicator of convex sets. It also identifies a broad class of subdifferentials with unique representations via explicit geometric data and a compatibility condition, and proves that in finite dimensions these assumptions can be relaxed. The results extend prior work on uniquely representable sublinear and indicator-function cases and motivate a conjecture that uniquely representable maximal monotone operators are either subdifferentials or linear skew-symmetric, thereby clarifying the landscape of operators with a singleton Fitzpatrick family.

Abstract

We study maximal monotone operators whose Fitzpatrick family reduces to a singleton; such operators will be called uniquely representable. We show that every such operator is cyclically monotone (hence, for some convex function ) if and only if it is 3-monotone. In Radon-Nikodým spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator is uniquely representable if and only if is the sum of a support and an indicator function of suitable convex sets.

Paper Structure

This paper contains 7 sections, 19 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Case of Maximal Monotone Operators
  4. Case of Subdifferential Operators
  5. Structural Properties of Subdifferentials with Unique Representative Function
  6. A class of subdifferentials with unique representative
  7. The Finite-Dimensional Case

Key Result

Lemma 3.1

For a maximal monotone operator $A:X \rightrightarrows X^*$ the following are equivalent: (i). $\mathcal{F}_A$ is a singleton, i.e. $\mathcal{F}_A=\{F_{A}\}$. (ii). $F_{A} \equiv P_{A}$. (iii). For every $(x,x^*),(y,y^*) \in X \times X^*$ If $A = \partial f$ for some proper convex lsc function $f:X \rightarrow \mathbb{R} \cup \{+\infty\}$, then the above assertions are also equivalent to the follo

Theorems & Definitions (42)

  • Lemma 3.1
  • proof
  • Proposition 3.2: Translation and Dilation Invariance
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 32 more