Convergence of the sparse Moment-SOS hierarchy for a 5-cycle quadratic instance

Determine whether the sequence of optimal values produced by the sparse Moment-SOS hierarchy (i.e., the k-th order sparse SOS/moment relaxations based on the correlative sparsity pattern) converges to the true optimum for the specific problem: minimize x1 x2 + x2 x3 + x3 x4 + x4 x5 + x5 x1 subject to x1 − x2 + x3 − x4 + x5 = 1, 2 x1 + x2 − x3 + 2 x4 − x5 ≥ 3, x1 + x2 + x3 + x4 + x5 = 5, and x ≥ 0. Equivalently, ascertain whether lim_{k→∞} f_k converges to f_min = 0 for this instance, where f_k denotes the optimal value of the k-th order sparse SOS (or moment) relaxation defined by the blockwise quadratic modules on the index sets {1,2}, {2,3}, {3,4}, {4,5}, and {5,1}.

Background

The paper introduces a new sparse Moment-SOS hierarchy tailored to sparse copositive polynomial optimization with possibly dense linear constraints. While dense Moment-SOS hierarchies have well-understood asymptotic convergence under Archimedean assumptions, convergence of sparse hierarchies in the presence of dense linear constraints is subtler and not guaranteed in general.

In the numerical Example ex:odd_cycle_5, the objective is the quadratic sum of adjacent products on a 5-cycle with linear equality and inequality constraints and nonnegativity. The authors observe that increasing the relaxation order yields progressively better lower bounds with the sparse hierarchy; however, they explicitly state uncertainty about whether these bounds converge to the true optimum for this instance. For comparison, the dense relaxation at order 1 is tight with f_min = 0 and the exact minimizer identified. This highlights a concrete unresolved question about convergence behavior of the proposed sparse hierarchy on this particular problem.