Moment/Sum-of-Squares Hierarchy for Complex Polynomial Optimization

Understand

We consider the problem of finding the global optimum of a real-valued complex polynomial on a compact set defined by real-valued complex polynomial inequalities.

• It reduces to solving a sequence of complex semidefinite programming relaxations that grow tighter and tighter thanks to D'Angelo's and Putinar's Positivstellenstatz discovered in 2008.
• In other words, the Lasserre hierarchy may be transposed to complex numbers.
• We propose a method for exploiting sparsity and apply the complex hierarchy to problems with several thousand complex variables.