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

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.