A combinatorial approach to Ramana's exact dual for semidefinite programming

Abstract

Thirty years ago, in a seminal paper Ramana derived an exact dual for Semidefinite Programming (SDP). Ramana's dual has the following remarkable features: i) it is an explicit, polynomial size semidefinite program ii) it does not assume that the primal is strictly feasible, nor does it make any other regularity assumptions iii) yet, it has strong duality with the primal. The complexity implications of Ramana's dual are fundamental, and to date still the best known. The most important of these is that SDP feasibility in the Turing model is not NP-complete, unless NP = co-NP. We give a treatment of Ramana's dual which is both simpler and more complete, than was previously available. First we connect it to a seemingly very different way of inducing strong duality: reformulating the SDP into a rank revealing form using elementary row operations and rotations. Second, while previous works characterized its objective value, we completely characterize its feasible set: in particular, we show it is a higher dimensional representation of an exact dual, which, however is not an explicit SDP. We also prove that -- somewhat surprisingly -- strict feasibility of Ramana's dual implies that the only feasible solution of the primal is the zero matrix. As a corollary, we obtain a short and transparent derivation of Ramana's dual, which we believe is accessible to both the optimization and the theoretical computer science communities. Our approach is combinatorial in the following sense: i) we use a minimum amount of continuous optimization theory ii) we show that feasible solutions in Ramana's dual are identified with regular facial reduction sequences, i.e., essentially discrete structures.