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

TL;DR

This paper extends the Lasserre hierarchy to complex polynomial optimization, enabling global solutions for large-scale power flow problems using complex semidefinite programming relaxations.

Contribution

It introduces a method to exploit sparsity in the complex hierarchy, allowing efficient optimization of large-scale problems with thousands of variables.

Findings

Successfully applied to power flow optimization in European transmission networks.

Achieved tighter relaxations through complex Positivstellenstatz.

Demonstrated scalability to problems with several thousand variables.

Abstract

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. These problems consist of computing optimal power flows in the European high-voltage transmission network.