Lower bounds on the global minimum of a polynomial

TL;DR

This paper introduces a new geometric programming-based lower bound for polynomial minima on bounded regions, which improves upon previous bounds and is computationally efficient for high-dimensional, sparse polynomials.

Contribution

It extends Ghasemi and Marshall's method to compute a lower bound on polynomial minima over bounded regions using geometric programming, outperforming existing semidefinite programming approaches in certain cases.

Findings

The new bound $f_{gp,M}$ improves upon $f_{gp}$ in numerical experiments.

Computations of $f_{gp,M}$ are faster and more scalable for large, sparse polynomials.

The method is practical for high-dimensional problems where semidefinite programming fails.

Abstract

We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$ on the closed ball ${x \in \mathbb{R}^n : \sum x_i^{2d} \le M}$, computable by geometric programming, for any real $M$. We compare this bound with the (global) lower bound $f_{{\rm gp}}$ obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796-816]. Our computations show that the bound $f_{{\rm gp},M}$ improves on the bound $f_{{\rm gp}}$ and that the computation of $f_{{\rm gp},M}$, like that of $f_{{\rm gp}}$, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down.