Exploiting Sparsity in Complex Polynomial Optimization
Jie Wang, Victor Magron
TL;DR
The paper develops sparsity-adapted complex moment-HSOS hierarchies for complex polynomial optimization problems (CPOPs) by exploiting correlative sparsity, term sparsity, and their combination. It provides theoretical convergence guarantees under maximal chordal extensions and introduces a minimum initial relaxation step to improve tightness and efficiency. Through extensive numerical experiments on randomly generated instances and AC-OPF problems, the authors demonstrate that the complex hierarchies can deliver competitive bounds at lower computational cost compared to the real hierarchy, often with substantial speedups. The work offers a self-contained framework for constructing and evaluating sparsity-aware complex relaxations, with practical impact on large-scale CPOPs in engineering domains such as power systems.
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.