Strengthened Complex Moment Hierarchies and Finite Convergence over Spheres

Abstract

This paper proposes strengthened complex moment relaxations for complex polynomial optimization. The proposed relaxation augments the pruned complex moment relaxation with selected positive semidefinite blocks from the full complex moment matrix. For the first strengthening, these blocks are motivated by the normality of multiplication operators and provide a tunable tradeoff between relaxation strength and computational cost through an additional normal order. In analogy with the real moment hierarchy, a flat-truncation condition for detecting global optimality is derived for the strengthened hierarchy. Furthermore, by considering the joint normality of multiplication operators, we propose a second strengthened complex moment hierarchy. For this stronger hierarchy, we are able to prove generic finite convergence and flat truncation for optimization over spheres. Numerical experiments demonstrate that the proposed strengthenings recover almost all the strength of the full relaxation at substantially lower computational cost.