Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

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 Rayleigh quotients is presented. Funding: F. Guo was supported by the Chinese National Natural Science Foundation [Grant 12471478]. J. Wang was funded by the National Key R&D Program of China [Grant 2023YFA1009401] and the Natural Science Foundation of China [Grants 12201618 and 12171324].