Finite convergence and minimizer extraction in moment relaxations with correlative sparsity

We identify a new sufficient condition for the finite convergence of moment relaxations of polynomial optimization problems with correlative sparsity. This condition, which follows from a solution to a correlatively sparse version of the classical truncated moment problem, requires that certain moment matrices admit a flat extension and that the variable cliques underpinning the relaxation satisfy a "running intersection" property. We also describe an algorithm that, when these conditions are met, extracts at least as many minimizers for the original polynomial optimization problem as the largest rank of the moment matrices in its relaxation. Our results, along with the necessity of the running intersection property, are illustrated with examples.