Generalisation of Farkas' lemma beyond closedness: a constructive approach via Fenchel-Rockafellar duality

TL;DR

This paper extends Farkas' lemma to cases where the cone's image isn't necessarily closed, using Fenchel-Rockafellar duality, under weaker assumptions than previous generalizations, and provides constructive conditions for approximate solutions.

Contribution

It introduces a new method to generalize Farkas' lemma without requiring the closedness of the cone's image, broadening its applicability in optimization.

Findings

Uncovers necessary and sufficient conditions for membership in the cone's image or its closure.

Provides constructive characterizations for approximate solutions with a given tolerance.

Discusses conditions for dual problem solvability and applications to nonconvex cones.

Abstract

Farkas' lemma is an ubiquitous tool in optimisation, as it provides necessary and sufficient conditions to have $b \in A(P)$, where $P$ is a closed convex cone, $A$ is a (continuous) linear mapping and $b$ is a fixed vector. The standard underlying hypothesis is the closedness of $A(P)$, which is not always satisfied and can be difficult to check. We devise a new method to generalise Farkas' lemma, based on a primal-dual pair of optimisation problems and Fenchel-Rockafellar duality theory. We work under the sole hypothesis that $P$ be generated by a closed bounded convex set. This hypothesis is weaker than in previous generalisations of Farkas' lemma, which almost all require that $A(P)$ be closed, or, in few cases, that only $P$ be closed. In our case, $P$ (and a fortiori $A(P)$) is not necessarily closed; we uncover necessary and sufficient conditions both for $b \in A(P)$ and $b \in \overline{A(P)}$. For a given $\e \geq 0$, we exhibit constructive characterisations of $x \in P$ such that $\|Ax-b\| \leq \e$ when it exists, by means of optimality conditions. For $\e = 0$, these strongly rely on whether the dual problem admits a solution, and we discuss conditions under which it does. Finally, we also explain how, upon relaxation, we may apply our method to a nonconvex cone.