Exact certification for conic optimization over quadratic modules: partitions of unity and recession directions

Abstract

We consider linear conic problems whose cone is a product of truncated quadratic modules, the sum-of-squares (SOS) side of the moment-SOS hierarchy. A floating-point solution is turned into an exact rational certificate in three steps: rounding, repair of each block residual by a constant, and absorption of the constants along a recession direction of the problem. The repair relies on strict partitions of unity in the quadratic modules. We characterize their existence, relate the canonical one to the analytic center of the dual pseudo-moment cone, Christoffel-Darboux polynomials and Pell equations, and we derive closed-form rational partitions for balls and boxes. The loss incurred by the certification is the optimal value of a recession problem, equal to a support function of the masses of the dual feasible pseudomoments. The method is illustrated on polynomial optimization, volume computation and positivity of polynomials along D-finite trajectories.