Solving SDP completely with an interior point oracle

We suppose the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying strong feasibility (i.e. 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 entails, 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 strong feasibility 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.

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