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Explicit data-dependent characterizations of the subdifferential of convex pointwise suprema and optimality conditions

Stephanie Caro, Abderrahim Hantoute

TL;DR

This work delivers explicit, data-dependent characterizations of the subdifferential and normal cone for the pointwise supremum of convex functions, expressed directly in terms of the underlying data functions. By symmetrically incorporating both $\varepsilon$-active and non-active components through data-driven $\varepsilon$-subdifferentials and $\varepsilon$-normal sets, it avoids domain-based geometric constructs and unifies previous results, including compact-continuous special cases. The authors derive sharp optimality conditions for infinite convex programs in terms of the data functions, yielding clear KKT/Fritz-John-type criteria that distinguish active and non-active constraints. Enhanced structural results under continuity assumptions provide further simplified representations, broadening applicability to infinite-dimensional and improper-function settings. Overall, the framework improves the tractability and interpretability of variational properties for supremum problems with potentially infinite constraints.

Abstract

We establish explicit data-dependent and symmetric characterizations of the subdifferential of the supremum of convex functions, formulated directly in terms of the underlying data functions. In our approach, both active and non-active functions contribute equally through their subdifferentials, thereby avoiding the need for additional geometric constructions, such as the domain of the supremum, that arise in previous developments. Applications to infinite convex optimization yield sharp Karush-Kuhn-Tucker and Fritz-John optimality conditions, expressed exclusively in terms of the objective and constraint functions and clearly distinguishing the roles of (almost) active and non-active constraints.

Explicit data-dependent characterizations of the subdifferential of convex pointwise suprema and optimality conditions

TL;DR

This work delivers explicit, data-dependent characterizations of the subdifferential and normal cone for the pointwise supremum of convex functions, expressed directly in terms of the underlying data functions. By symmetrically incorporating both -active and non-active components through data-driven -subdifferentials and -normal sets, it avoids domain-based geometric constructs and unifies previous results, including compact-continuous special cases. The authors derive sharp optimality conditions for infinite convex programs in terms of the data functions, yielding clear KKT/Fritz-John-type criteria that distinguish active and non-active constraints. Enhanced structural results under continuity assumptions provide further simplified representations, broadening applicability to infinite-dimensional and improper-function settings. Overall, the framework improves the tractability and interpretability of variational properties for supremum problems with potentially infinite constraints.

Abstract

We establish explicit data-dependent and symmetric characterizations of the subdifferential of the supremum of convex functions, formulated directly in terms of the underlying data functions. In our approach, both active and non-active functions contribute equally through their subdifferentials, thereby avoiding the need for additional geometric constructions, such as the domain of the supremum, that arise in previous developments. Applications to infinite convex optimization yield sharp Karush-Kuhn-Tucker and Fritz-John optimality conditions, expressed exclusively in terms of the objective and constraint functions and clearly distinguishing the roles of (almost) active and non-active constraints.
Paper Structure (7 sections, 18 theorems, 131 equations)

This paper contains 7 sections, 18 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Characterization of the normal cone to the domain
  4. Subdifferential of the supremum function
  5. Optimality conditions
  6. Enhanced structural framework
  7. Concluding remarks

Key Result

Lemma 1

Let $f_{t}:X\rightarrow\overline{\mathbb{R}}$, $t\in T,$ be lsc convex functions, and let $A\subset X$ be a convex set.$\ $Fix $\varepsilon>0.$ Then, for all $\delta>0$ and $\beta_{t}>0$ with $t\in\mathcal{P}:=\{t\in T:f_{t}$ is proper$\},$ Consequently, for any scalars $\alpha_{t},\beta_{t}>0$$(t\in\mathcal{P})$ such that $\sup_{t\in\mathcal{P}}\alpha_{t}\beta_{t}^{-1}<+\infty$ and $\sup_{t\in\ma

Theorems & Definitions (26)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Remark 1: before the proof
  • Corollary 7
  • Corollary 8
  • ...and 16 more