Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

• 成果类型: Article; Early Access
• 署名作者: Guo, Feng; Wang, Jie; Zheng, Jianhao
• 署名单位: Dalian University of Technology; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2024.0541
• 发表日期: 2025-12-01
• 关键词: sum of rational functions; sign symmetry; semidefinite relaxation; correlative sparsity; generalized Rayleigh quotient; JACOBIAN SDP RELAXATION; global optimization; hierarchy

摘要

This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum-of-squares (SOS) relaxations that is dual to the generalized moment problem approach proposed by Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: (1) it allows us to conduct a convergence rate analysis for the hierarchy; (2) it leads to a sign symmetry-adapted hierarchy consisting of block-diagonal semidefinite relaxations. When the problem possesses correlative sparsity as well as sign symmetries, we propose sparse semidefinite relaxations by exploiting both structures. Various numerical experiments are performed to demonstrate the efficiency of our approach. Finally, an application to maximizing sums of generalized Rayis