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A Bilevel Hierarchy of Strengthened Complex Moment Relaxations for Complex Polynomial Optimization

Jie Wang

TL;DR

This work tackles slow convergence in complex polynomial optimization by introducing a bilevel strengthened complex moment hierarchy that leverages normality-inspired PSD constraints tied to a second-order parameter $s$. The approach yields a family of relaxations $tauPrime_{r,s}$ that tighten bounds without increasing the outer order and can be integrated with correlative sparsity for scalability. The authors establish a theoretical link between these constraints and the normality of shift operators, provide monotonicity and convergence insights under sphere constraints, and demonstrate substantial empirical gains across quadratic, quartic, and conjecture-related problems. Overall, the method delivers tighter bounds at lower computational cost, broadening the practical applicability of complex moment relaxations in areas like signal processing, power systems, quantum information, and combinatorial optimization.

Abstract

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.

A Bilevel Hierarchy of Strengthened Complex Moment Relaxations for Complex Polynomial Optimization

TL;DR

This work tackles slow convergence in complex polynomial optimization by introducing a bilevel strengthened complex moment hierarchy that leverages normality-inspired PSD constraints tied to a second-order parameter . The approach yields a family of relaxations that tighten bounds without increasing the outer order and can be integrated with correlative sparsity for scalability. The authors establish a theoretical link between these constraints and the normality of shift operators, provide monotonicity and convergence insights under sphere constraints, and demonstrate substantial empirical gains across quadratic, quartic, and conjecture-related problems. Overall, the method delivers tighter bounds at lower computational cost, broadening the practical applicability of complex moment relaxations in areas like signal processing, power systems, quantum information, and combinatorial optimization.

Abstract

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.
Paper Structure (13 sections, 4 theorems, 26 equations, 6 tables)

This paper contains 13 sections, 4 theorems, 26 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The main results
  4. Strengthening the complex moment relaxation
  5. Link to the normality of shift operators
  6. Integrating with correlative sparsity
  7. Numerical experiments
  8. Minimizing a random complex quadratic polynomial with unit-norm variables
  9. Minimizing a random complex quartic polynomial on a sphere
  10. Minimizing a random complex quartic polynomial with correlative sparsity on multi-spheres
  11. Smale’s Mean Value conjecture
  12. The Mordell inequality conjecture
  13. Discussions

Key Result

Proposition 3.1

\newlabelsec4:thm10 Suppose that ${\mathbf{y}}$ is a complex sequence that admits a finitely atomic representing measure. Then for $i\in[n]$ and $s\in{\mathbb{N}}^*$, it holds

Theorems & Definitions (16)

  • Remark 2.1
  • Proposition 3.1
  • Proof 1
  • Remark 3.2
  • Theorem 3.3
  • Proof 2
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 6 more