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.
Younes Naceur, Jie Wang, Victor Magron, Antonio Acín
Semidefinite relaxations of polynomial optimization have become a central tool for addressing the non-convex optimization problems over non-commutative operators that are ubiquitous in quantum information theory and, more in general, quantum physics. Yet, as these global relaxation methods are solved numerically, the bounds issued by the semidefinite solver can - and often do - exceed the global optimum, undermining their certifiability. To counter this issue, we introduce a framework for extracting rigorous certified bounds on non-commutative optimization problems from inexact numerical data, and apply it to several paradigmatic problems in quantum information theory. An extension to sparsity and symmetry-adapted semidefinite relaxations is also provided and compared to the general dense scheme. Our results establish rational post-processing as a practical route to reliable certification, pushing semidefinite optimization toward a certifiable standard for quantum information science.