Exact Duality in Semidefinite Programming Based on Elementary Reformulations

In semidefinite programming (SDP), unlike in linear programming, Farkas’ lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any equality constrained semidefinite system using only elementary row operations, and rotations. When a system is infeasible, the reformulated system is trivially infeasible. When a system is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of all infeasible SDPs and the constraints of all feasible SDPs with a fixed rank maximal solution. Our elementary reformulations can be constructed either by a direct method, or by adapting the Waki-Muramatsu facial reduction algorithm. In different language, the reformulations provide a standard form of spectrahedra, to easily verify either their emptiness, or a tight upper bound on the rank of feasible solutions.