A symbolic algorithm for exact polynomial optimization strengthened with Fritz John conditions

TL;DR

This paper introduces a symbolic algorithm that leverages Fritz John conditions to exactly compute the optimal value of polynomial optimization problems, utilizing real radical generators and Gr"obner basis techniques.

Contribution

The paper presents a novel symbolic algorithm that incorporates Fritz John conditions to enhance the exact computation of polynomial optimization solutions.

Findings

Algorithm computes exact optimal values for polynomial problems.

Method applies to problems with complementarity constraints.

Utilizes real radical generators and Gr"obner basis for computation.

Abstract

Consider a polynomial optimization problem. Adding polynomial equations generated by the Fritz John conditions to the constraint set does not change the optimal value. As proved in [arXiv:2205.04254 (2022)], the objective polynomial has finitely many values on the new constraint set under some genericity assumption. Based on this, we provide an algorithm that allows us to compute exactly this optimal value. Our method depends on the computations of real radical generators and Gr\"obner basis. Finally, we apply our method to solve some instances of mathematical program with complementarity constraints.