Minimizing Rational Functions by Exact Jacobian SDP Relaxation Applicable to Finite Singularities

TL;DR

This paper introduces an exact Jacobian SDP relaxation approach for minimizing rational functions by reformulating them as polynomial optimization problems, demonstrating efficiency and extending applicability to cases with finite singularities.

Contribution

It extends Nie's Jacobian SDP relaxation method to handle rational functions with finite singularities, providing a more general and efficient solution approach.

Findings

Method effectively minimizes rational functions.

Reformulation as polynomial optimization is equivalent under generic conditions.

Numerical examples demonstrate efficiency and applicability.

Abstract

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as polynomial optimization by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization. We also prove that the assumption of nonsingularity in Nie's method can be weakened as the finiteness of singularities. Some numerical examples are given to illustrate the efficiency of our method.