  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 due to 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.
