This paper proposes a bilevel hierarchy of strengthened complex moment relaxations for complex polynomial optimization.
The key trick entails considering a class of positive semidefinite conditions that arise naturally in characterizing the normality of the so-called shift operators. The relaxation problem in this new hierarchy is parameterized by the usual relaxation order as well as an extra normal order, thus providing more space of flexibility to balance the strength of relaxation and computational complexity.
Extensive numerical experiments demonstrate the superior performance of the new hierarchy compared to the usual hierarchy.