Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as a polynomial optimization problem 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 problem. We also prove that the assumption of nonsingularity in Nie’s method can be weakened to the finiteness of singularities. Some numerical examples are given in the end.