Unconstraint global polynomial optimization via Gradient Ideal

TL;DR

This paper introduces a novel finite-step method for global polynomial optimization that combines Border Basis, Moment Matrices, and Semidefinite Programming to find minima and their defining ideals, assuming zero-dimensional minimizer ideals.

Contribution

It generalizes Lasserre relaxation, providing a finite-step algorithm for computing polynomial minima and minimizer ideals using an integrated approach.

Findings

Successfully computes polynomial minima with finite steps

Provides border basis of the minimizer ideal when minima are finite

Extends existing relaxation methods to broader classes of problems

Abstract

In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function, assuming that the minimizer ideal is zero-dimensional. Our method is a generalization of Lasserre relaxation method and stops in a finite number of steps. The proposed algorithm combines Border Basis, Moment Matrices and Semidefinite Programming. In the case where the minimum is reached at a finite number of points, it provides a border basis of the minimizer ideal.