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Relaxation in infinite convex programming under Slater-type regularity conditions

Rafael Correa, Abderrahim Hantoute, Marco A. López

TL;DR

This work analyzes zero-duality-gap conditions for infinite convex programs via biconjugate relaxations. It shows that Slater-type and continuity-type conditions yield no gap between the original problem and its standard biconjugate relaxation in finite-constraint settings, and introduces a reinforced relaxation based on $f_{\infty}=\sup_k f_k$ to handle infinitely many constraints, achieving zero gap under mild regularity. The authors connect these relaxations to Fenchel duality, demonstrating that zero gaps propagate through Fenchel-type duals and that supremum-interchangeability plays a central role. They also provide a concrete nonreflexive-space counterexample where the standard biconjugate relaxation fails, thereby motivating the reinforced approach and highlighting the practical impact for duality theory in infinite convex optimization.

Abstract

The main purpose of this paper is to close the gap between the optimal values of an infinite convex program and that of its biconjugate relaxation. It is shown that Slater and continuity-type conditions guarantee such a zero-duality gap. The approach uses calculus rules for the conjugation and biconjugation of the sum and pointwise supremum operations. A second important objective of this work is to exploit these results on relaxation by applying them in the context of duality theory.

Relaxation in infinite convex programming under Slater-type regularity conditions

TL;DR

This work analyzes zero-duality-gap conditions for infinite convex programs via biconjugate relaxations. It shows that Slater-type and continuity-type conditions yield no gap between the original problem and its standard biconjugate relaxation in finite-constraint settings, and introduces a reinforced relaxation based on to handle infinitely many constraints, achieving zero gap under mild regularity. The authors connect these relaxations to Fenchel duality, demonstrating that zero gaps propagate through Fenchel-type duals and that supremum-interchangeability plays a central role. They also provide a concrete nonreflexive-space counterexample where the standard biconjugate relaxation fails, thereby motivating the reinforced approach and highlighting the practical impact for duality theory in infinite convex optimization.

Abstract

The main purpose of this paper is to close the gap between the optimal values of an infinite convex program and that of its biconjugate relaxation. It is shown that Slater and continuity-type conditions guarantee such a zero-duality gap. The approach uses calculus rules for the conjugation and biconjugation of the sum and pointwise supremum operations. A second important objective of this work is to exploit these results on relaxation by applying them in the context of duality theory.
Paper Structure (7 sections, 12 theorems, 133 equations)

This paper contains 7 sections, 12 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Relaxation theory
  4. Concave-like setting
  5. Relationship with duality in convex programming
  6. Biconjugate relaxation $(\mathcal{P}^{\ast\ast})$ is not enough
  7. Conclusions

Key Result

Proposition 1

Assume that the convex functions $f_{k},$$k=1,\cdots,m$, are finite and continuous at some point in $\operatorname*{dom}f_{0}.$ Then, under the Slater condition, we have

Theorems & Definitions (19)

  • Remark 1
  • Definition 1
  • Proposition 1
  • Proposition 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Corollary 7
  • Corollary 8
  • ...and 9 more