Lower bounds for polynomials using geometric programming

TL;DR

This paper introduces a new, efficiently computable lower bound for polynomials using geometric programming, generalizing previous sum of squares bounds and offering faster computation at the expense of some tightness.

Contribution

It presents a novel sufficient condition for polynomials to be sums of squares and derives a new lower bound via geometric programming, extending prior results.

Findings

The geometric programming-based lower bound is faster to compute than semidefinite programming bounds.

The new lower bound generalizes previous results and applies to polynomials with specific highest degree terms.

Computational experiments show practical efficiency of the geometric programming approach.

Abstract

We make use of a result of Hurwitz and Reznick, and a consequence of this result due to Fidalgo and Kovacec, to determine a new sufficient condition for a polynomial $f\in\mathbb{R}[X_1,...,X_n]$ of even degree to be a sum of squares. This result generalizes a result of Lasserre and a result of Fidalgo and Kovacec, and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound $f_{gp}$ for $f$, and we explain how $f_{gp}$ can be computed using geometric programming. The lower bound $f_{gp}$ is generally not as good as the lower bound $f_{sos}$ introduced by Lasserre and Parrilo and Sturmfels, which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of $f_{gp}$ is much faster. The computation is simplest when the highest degree term of $f$ has the form $\sum_{i=1}^n a_iX_i^{2d}$, $a_i>0$, $i=1,...,n$. The lower bounds for $f$ established in [6] are obtained by evaluating the objective function of the geometric program at the appropriate feasible points.