Algebraic Farkas Lemma and Strong Duality for Perturbed Conic Linear Programming
P. D. Khanh, V. V. H. Khoa, T. H. Mo
TL;DR
The paper develops an algebraic theory of Farkas lemma and strong duality for infinite-dimensional conic linear programs in dual pairs of vector spaces by analyzing perturbed value functions $v_{\mathcal{D}}$ and $v_{\mathcal{P}}$ and their epigraph/hypograph sets $\mathcal{N}$ and $\mathcal{M}$. It introduces hypergraphical/epigraphical constructions inspired by Kretschmer's closedness conditions and derives perturbed Farkas lemmas that characterize zero duality gap, linking them to perturbed strong duality. The authors provide algebraic sufficient conditions via the algebraic core of $P^*$ and, under topological augmentation, precise topological characterizations showing $\mathcal{N}$ and $\mathcal{M}$ as closures of $\mathcal{H}$ and $\mathcal{K}$. This yields a unified, purely algebraic pathway to strong duality and duality-gap analysis that can be completed with topological assumptions when available, broadening the applicability of duality theory in very general spaces.
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.