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Variational properties of the abstract subdifferential operator

Reinier Diàz Millàn, Nadezda Sukhorukova, Julien Ugon

TL;DR

This work extends convex analysis to abstract convexity defined by a function family $L$, focusing on the abstract subdifferential $\partial_L f$ and its calculus. It develops sum and composition rules without relying on classical Minkowski separation theorems, and proves that under a mild boundedness assumption the subdifferential is a maximal abstract monotone operator. A key contribution is a counterexample showing that two disjoint $L$-convex sets may not be separable by an abstract affine function, underscoring a fundamental obstacle for abstract-convexity-based numerics. The results lay a theoretical foundation for numerical methods in abstract convexity and indicate open avenues in separation theory and discrete optimization.

Abstract

Abstract convexity generalises classical convexity by considering the suprema of functions taken from an arbitrarily defined set of functions. These are called the abstract linear (abstract affine) functions. The purpose of this paper is to study the abstract subdifferential. We obtain a number of results on the calculus of this subdifferential: summation and composition rules, and prove that under some reasonable conditions the subdifferential is a maximal abstract monotone operator. Another contribution of this paper is a counterexample that demonstrates that the separation theorem between two abstract convex sets is generally not true. The lack of the extension of separation results to the case of abstract convexity is one of the obstacles in the development of abstract convexity based numerical methods.

Variational properties of the abstract subdifferential operator

TL;DR

This work extends convex analysis to abstract convexity defined by a function family , focusing on the abstract subdifferential and its calculus. It develops sum and composition rules without relying on classical Minkowski separation theorems, and proves that under a mild boundedness assumption the subdifferential is a maximal abstract monotone operator. A key contribution is a counterexample showing that two disjoint -convex sets may not be separable by an abstract affine function, underscoring a fundamental obstacle for abstract-convexity-based numerics. The results lay a theoretical foundation for numerical methods in abstract convexity and indicate open avenues in separation theory and discrete optimization.

Abstract

Abstract convexity generalises classical convexity by considering the suprema of functions taken from an arbitrarily defined set of functions. These are called the abstract linear (abstract affine) functions. The purpose of this paper is to study the abstract subdifferential. We obtain a number of results on the calculus of this subdifferential: summation and composition rules, and prove that under some reasonable conditions the subdifferential is a maximal abstract monotone operator. Another contribution of this paper is a counterexample that demonstrates that the separation theorem between two abstract convex sets is generally not true. The lack of the extension of separation results to the case of abstract convexity is one of the obstacles in the development of abstract convexity based numerical methods.
Paper Structure (7 sections, 27 theorems, 55 equations, 2 figures)

This paper contains 7 sections, 27 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Abstract Conjugate
  5. Abstract Subdifferential Calculus
  6. The abstract subdifferential as an abstract monotone operator
  7. Conclusions and future research directions

Key Result

Proposition 1

Let $A_i: i\in I$ be a family of $H$-convex sets. Then $\cap_{i\in I}A_i$ is also $H$-convex.

Figures (2)

  • Figure 1: The sets $A$ and $B$ are both $H$-convex and disjoint, but they cannot be separated by a point $x$.
  • Figure 2: The function $u$ is a $L_1$-subgradient of the function $|x|$ at $x=1$.

Theorems & Definitions (68)

  • Definition 1: Abstract Convexity rubinov:2000
  • Definition 2: Support Set rubinov:2000
  • Proposition 1: diaz-millan.ea:2023
  • Definition 3: Abstract Convex Hull of a function rubinov:2000
  • Definition 4: Abstract Convex Hull of a set rubinov:2000
  • Proposition 2: rubinov:2000
  • Definition 5: $(X,L)$-convex set
  • Definition 6: $(X,L)$-convex function
  • Definition 7: Support set, dual version
  • Proposition 3
  • ...and 58 more