Minimizing the sum of many rational functions

TL;DR

This paper presents a method to globally minimize the sum of multiple rational functions over semialgebraic sets using a hierarchy of convex relaxations, applicable to large-scale problems with sparsity.

Contribution

It formulates the rational optimization as a generalized moment problem and demonstrates convergence of the relaxation hierarchy to the global optimum.

Findings

Hierarchy of semidefinite relaxations converges under certain conditions.

Method effectively handles large problems with sparsity.

Public software can be used for modeling and solving these problems.

Abstract

We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of variables can be large (10 to 100) provided some kind of sparsity is present. We describe a formulation of the rational optimization problem as a generalized moment problem and its hierarchy of convex semidefinite relaxations. Under some conditions we prove that the sequence of optimal values converges to the globally optimal value. We show how public-domain software can be used to model and solve such problems.