Solving SDP Completely with an Interior Point Oracle

TL;DR

This paper demonstrates how an oracle capable of solving well-conditioned SDPs can be used to fully solve any SDP, including feasibility, optimality, and infeasibility detection, through advanced facial reduction techniques.

Contribution

It introduces a method to leverage an SDP oracle to completely solve arbitrary SDPs by employing double facial reduction and Slater's condition.

Findings

The approach can distinguish feasibility and infeasibility cases.

It can determine whether the optimal value is attained.

The techniques extend to general convex cones.

Abstract

We suppose the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying Slater's condition simultaneously at its primal and dual sides. We note that such an oracle might not be able to directly solve general SDPs even after certain regularization schemes are applied. In this work we fill this gap and show how to use such an oracle to "completely solve" an arbitrary SDP. Completely solving an SDP, includes, for example, distinguishing between weak/strong feasibility/infeasibility and detecting when the optimal value is attained or not. We will employ several tools, including a variant of facial reduction where all auxiliary problems are ensured to satisfy Slater's condition at all sides. Our main technical innovation, however, is an analysis of double facial reduction, which is the process of applying facial reduction twice: first to the original problem and then once more to the dual of the regularized problem obtained during the first run. Although our discussion is focused on semidefinite programming, the majority of the results are proved for general convex cones