Conic programming: infeasibility certificates and projective geometry

TL;DR

This paper introduces a projective geometry approach to conic programming that provides polynomial-time infeasibility certificates and distinguishes stably infeasible problems, improving understanding and verification of infeasibility.

Contribution

It presents a homogenization strategy based on projective geometry that eliminates weak infeasibility and offers new certificates for semidefinite programs.

Findings

Infeasibility certificates can be checked in polynomial time.

Homogenization eliminates weak infeasibility.

Stably infeasible problems have rational certificates.

Abstract

We revisit facial reduction from the point of view of projective geometry. This leads us to a homogenization strategy in conic programming that eliminates the phenomenon of weak infeasibility. For semidefinite programs (and others), this yields infeasibility certificates that can be checked in polynomial time. Furthermore, we propose a refined type of infeasibility, which we call stably infeasible, for which rational infeasibility certificates exist and that can be distinguished from other infeasibility types by our homogenization.