Exploiting term sparsity in Noncommutative Polynomial Optimization

Abstract: We provide a new hierarchy of semidefinite programming relaxations, called NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the exploitation of term sparsity hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits correlative sparsity for noncommutative optimization problems by Klep, Magron and Povh in arXiv:1909.00569, and is the noncommutative analogue of the TSSOS framework by Wang, Magron and Lasserre in arXiv:1912.08899. We also propose an extension exploiting simultaneously correlative and term sparsity, as done previously in the commutative case arXiv:2005.02828. Under certain conditions, we prove that the optimums of the NCTSSOS hierarchy converge to the optimum of the corresponding dense SDP relaxation. We illustrate the efficiency and scalability of NCTSSOS by solving eigenvalue/trace optimization problems from the literature as well as randomly generated examples involving up to several thousands of variables.