Relaxed Lagrangian duality in convex infinite optimization: reverse strong duality and optimality

TL;DR

This paper develops a duality framework for convex infinite optimization problems, establishing reverse strong duality, Farkas lemmas, and optimality conditions, especially for infinite and semi-infinite linear cases.

Contribution

It introduces a Lagrangian-Haar duality approach for convex problems on locally convex spaces with infinite index sets, extending duality theory.

Findings

Established reverse H-strong duality theorems

Proved H-Farkas type lemmas for infinite problems

Derived optimality theorems for semi-infinite linear optimization

Abstract

We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.