Relaxation in infinite convex programming under Slater-type regularity conditions

TL;DR

This paper investigates the conditions under which the optimal values of infinite convex programs and their relaxations coincide, using Slater-type regularity conditions to establish zero-duality gaps and applying conjugation calculus.

Contribution

It provides new conditions ensuring zero-duality gap in infinite convex programming and applies conjugation calculus to analyze relaxation and duality.

Findings

Slater and continuity conditions guarantee zero-duality gap.

Conjugation calculus rules are effective for analyzing relaxation.

Results enhance understanding of duality 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.