Chordal-TSSOS: A Moment-SOS Hierarchy That Exploits Term Sparsity with Chordal Extension
Jie Wang, Victor Magron, Jean B. Lasserre
Abstract
This work is a follow-up and a complement to [J. Wang, V. Magron and J. B. Lasserre, preprint, arXiv:1912.08899, 2019] where the TSSOS hierarchy was proposed for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose is a new sparse moment-SOS framework based on term sparsity and chordal extension. By exploiting term sparsity of the input polynomials we obtain a two-level hierarchy of semidefinite programming relaxations. The novelty and distinguishing feature of such relaxations is to involve block matrices obtained in an iterative procedure that performs chordal extension of certain adjacency graphs. The graphs are related to the terms arising in the original data and not to the links between variables. Various numerical examples demonstrate the efficiency and the scalability of this new hierarchy for both unconstrained and constrained POPs. The two hierarchies are complementary. While the former TSSOS [J. Wang, V. Magron and J. B. Lasserre, preprint, arXiv:1912.08899, 2019] has a theoretical convergence guarantee (to the dense moment-SOS relaxation), the chordal-TSSOS has superior performance but lacks this theoretical guarantee.