An experimental approach for global polynomial optimization based on Moments and Semidefinite Programming

TL;DR

This paper introduces an experimental algorithm that computes upper bounds for the global minimum of real polynomials using moment relaxations and semidefinite programming, with applications to nonnegative polynomials and gradient variety minimization.

Contribution

It presents a novel algorithm that enhances moment relaxation methods by incorporating flatness conditions to better approximate polynomial global minima.

Findings

Algorithm provides tight upper bounds in many cases.

Effective for nonnegative polynomials not expressible as sums of squares.

Numerical results demonstrate applicability to gradient variety minimization.

Abstract

In this article we provide an experimental algorithm that in many cases gives us an upper bound of the global infimum of a real polynomial on $\R^{n}$. It is very well known that to find the global infimum of a real polynomial on $\R^{n}$, often reduces to solve a hierarchy of positive semidefinite programs, called moment relaxations. The algorithm that we present involves to solve a series of positive semidefinite programs whose feasible set is included in the feasible set of a moment relaxation. Our additional constraint try to provoke a flatness condition, like used by Curto and Fialkow, for the computed moments. At the end we present numerical results of the application of the algorithm to nonnegative polynomials which are not sums of squares. We also provide numerical results for the application of a version of the algorithm based on the method proposed by Nie, Demmel and Sturmfels for the problem of minimizing a polynomial over its gradient variety.