Leveraging Christoffel-Darboux Kernels to Strengthen Moment-SOS Relaxations
The classical Moment-SOS hierarchy allows one to approximate the global minimum of a polynomial optimization problem through semidefinite relaxations of increasing accuracy. For many optimization problems, however, moving to higher relaxation orders quickly becomes computationally prohibitive. While this difficulty is commonly addressed by exploiting structural properties such as symmetry or sparsity, we explore a complementary direction: strengthening a low-order relaxation using information contained in its own pseudo-moment solution. To this end, we leverage Christoffel-Darboux kernels to identify regions that are compatible with the current pseudo-moment information and use them to construct additional polynomial constraints. We propose two mechanisms: the first, denoted $\HH$, is based on successive global Christoffel restrictions, while the second, denoted $\hh$, combines marginal Christoffel information with an available local solution. For $\HH$, we establish a uniform separation property for successive pseudo-moment solutions and geometric convergence of the fixed-order bounds on a particular subclass of knapsack problems. Moreover, for specific Boolean quadratic problems, we derive an explicit $\hh$ selection rule and establish conditions under which the resulting marginal constraints preserve all global minimizers. Numerical experiments on dense and sparse nonconvex quadratically constrained quadratic programs show that the resulting strengthened low-order relaxations can significantly improve the initial bounds, sometimes attaining exactness, while remaining substantially cheaper than solving higher-order relaxations.