Sparse noncommutative polynomial optimization

• 成果类型: Article
• 署名作者: Klep, Igor; Magron, Victor; Povh, Janez
• 署名单位: University of Ljubljana; Centre National de la Recherche Scientifique (CNRS); University of Ljubljana
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01610-1
• 发表日期: 2022
• 页码: 789-829
• 关键词: positive polynomials; sdp-relaxations; sums; squares; bounds; positivstellensatz; factorization; minimization; symmetries; forms

摘要

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822-843, 2006) and Waki et al. (SIAM J Optim 17(1):218-242, 2006). The Gelfand-Naimark-Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.