2020/10/14 by Jie Wang, Victor Magron, Wang, Jie +1 · 2 citations
Computer Science · Engineering · Mathematics · #12D15 #14P10 #47N10 #90C22 #Advanced Optimization Algorithms Research #Algebra over a field #Commutative property #Complexity and Algorithms in Graphs #Computer science #Correlative #Discrete mathematics #Eigenvalues and eigenvectors #Extension (predicate logic) #FOS: Mathematics #Hierarchy #Mathematical optimization #Mathematics #Noncommutative geometry #Optimization and Control (math.OC) #Optimization problem #Polynomial #Primary #Pure mathematics #Relaxation (psychology) #Scalability #Secondary #Semidefinite programming #Sparse and Compressive Sensing Techniques #TRACE (psycholinguistics) #Term (time) #Theoretical computer science #math.OC #msc:12D15 #msc:14P10 #msc:47N10 #msc:90C22
paper · pdf · doi:10.48550/arxiv.2010.06956
published in arXiv (Cornell University) (Cornell University) · 33 pages, 5 figures, 12 tables
arxiv created 2020/10/14 · openalex publication_date 2020/10/14 · arxiv updated 2020/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.