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

Jie Wang, Victor Magron

TL;DR

It is proved 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.

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.

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

TL;DR

It is proved 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.

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.
Paper Structure (8 sections, 8 theorems, 55 equations, 8 tables)

This paper contains 8 sections, 8 theorems, 55 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The complex moment-HSOS hierarchy
  4. The real moment-HSOS hierarchy
  5. Detecting global optimality and extracting solutions
  6. Numerical experiments
  7. Application in polyphase code design
  8. Conclusions

Key Result

Theorem 3.1

\newlabelthm10 Let $f=\sum_{({\boldsymbol{\beta}},{\boldsymbol{\gamma}})}f_{{\boldsymbol{\beta}},{\boldsymbol{\gamma}}}{\mathbf{z}}^{{\boldsymbol{\beta}}}\overline{{\mathbf{z}}}^{{\boldsymbol{\gamma}}}\in{\mathbb{R}}[{\mathbf{z}},\overline{{\mathbf{z}}}]^{\rm{c}}$. Then it holds

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2
  • Theorem 3.3
  • Proof 3
  • Remark 3.4
  • Corollary 4.1
  • Proof 4
  • ...and 10 more