A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

TL;DR

This paper introduces a real moment-HSOS hierarchy for complex polynomial optimization with real coefficients, offering computational advantages while maintaining solution accuracy, and provides conditions for global optimality and solution extraction.

Contribution

The paper develops a real hierarchy that matches the complex hierarchy's bounds but is computationally cheaper, and proves conditions for global optimality and solution extraction.

Findings

Real hierarchy achieves the same bounds as the complex hierarchy.

Global optimality is guaranteed under certain rank conditions.

Numerical examples demonstrate efficiency and practical application.

Abstract

This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to solve. In addition, we prove that global optimality is achieved when the ranks of the moment matrix and certain submatrix equal two in case that a sphere constraint is present, and as a consequence, the complex polynomial optimization problem has either two real optimal solutions or a pair of conjugate optimal solutions. A simple procedure for extracting a pair of conjugate optimal solutions is given in the latter case. Various numerical examples are presented to demonstrate the efficiency of this new hierarchy, and an application to polyphase code design is also provided.