Algebraic Farkas Lemma and Strong Duality for Perturbed Conic Linear Programming

TL;DR

This paper develops algebraic Farkas lemmas and strong duality results for infinite-dimensional conic linear programming, using hypergraphical and epigraphical sets to characterize zero duality gap.

Contribution

It introduces new algebraic and convex separation conditions for perturbed optimal value functions in infinite-dimensional conic programming, extending duality theory.

Findings

Established algebraic Farkas lemmas for infinite-dimensional cones

Provided characterizations of zero duality gap under algebraic and topological conditions

Developed hypergraphical and epigraphical set frameworks for duality analysis

Abstract

This paper addresses the study of algebraic versions of Farkas lemma and strong duality results in the very broad setting of infinite-dimensional conic linear programming in dual pairs of vector spaces. To this end, purely algebraic properties of perturbed optimal value functions of both primal and dual problems and their corresponding hypergraph/epigraph are investigated. The newly developed hypergraphical/epigraphical sets, inspired by Kretschmer's closedness conditions \cite{Kretschmer61}, together with their novel convex separation-type characterizations, give rise to various perturbed Farkas-type lemmas which allow us to derive complete characterizations of ``zero duality gap''. Principally, when certain structures of algebraic or topological duals are imposed, illuminating implications of the developed condition are also explored.