Exploiting Sparsity in Complex Polynomial Optimization

TL;DR

This paper introduces a sparsity-adapted hierarchy for complex polynomial optimization that balances computational efficiency and solution quality, especially for large-scale problems.

Contribution

It develops and compares a complex moment-HSOS hierarchy with real counterparts, demonstrating advantages in large-scale complex optimization tasks.

Findings

The complex hierarchy offers better computational efficiency.

It provides tighter bounds than real hierarchies in large problems.

Numerical results validate the approach's effectiveness.

Abstract

In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierarchy on either randomly generated complex polynomial optimization problems or the AC optimal power flow problem. The results of numerical experiments show that the sparsity-adapted complex moment-HSOS hierarchy provides a trade-off between the computational cost and the quality of obtained bounds for large-scale complex polynomial optimization problems.