Strong duality in infinite convex optimization

TL;DR

This paper introduces new dual formulations for infinite convex optimization that ensure zero duality gap and strong duality under standard conditions, expanding the theoretical understanding of infinite-dimensional problems.

Contribution

It develops novel Lagrangian dual formulations involving infinite variables and sums, guaranteeing strong duality where classical methods may not.

Findings

New dual formulations achieve zero duality gap.

Strong duality holds under standard Slater condition.

Provides general optimality conditions for infinite sums.

Abstract

We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums of functions. Unlike the classical Haar duality scheme, these dual problems provide zero duality gap and are solvable under the standard Slater condition. Then we derive general optimality conditions/multiplier rules by applying subdifferential rules for infinite sums established in [13].