Relaxed Lagrangian duality in convex infinite optimization: reducibility and strong duality

TL;DR

This paper introduces a new duality framework for convex infinite optimization problems, providing conditions for reducibility and strong duality, with applications to linear and semi-infinite problems.

Contribution

It develops a Lagrangian-Haar duality approach for convex problems with infinitely many constraints, establishing necessary and sufficient conditions for reducibility and strong duality.

Findings

Conditions for H-reducibility and equivalence to subproblems

Zero H-duality gap and H-stable strong duality results

Applications to linear and semi-infinite convex optimization

Abstract

We associate with each convex optimization problem, posed on some locally convex space, with infinitely many constraints indexed by the set T, and a given non-empty family H of finite subsets of T, a suitable Lagrangian-Haar dual problem. We obtain necessary and sufficient conditions for H-reducibility, that is, equivalence to some subproblem obtained by replacing the whole index set T by some element of H. Special attention is addressed to linear optimization, infinite and semi-infinite, and to convex problems with a countable family of constraints. Results on zero H-duality gap and on H-(stable) strong duality are provided. Examples are given along the paper to illustrate the meaning of the results.