Linear Optimization of Polynomials and Rational Functions over Boxes

TL;DR

This paper introduces a method for deriving tight linear lower bounds for multivariate polynomials and rational functions over boxes, utilizing Bernstein form and degree elevation, with convergence analysis and numerical comparisons.

Contribution

It presents a novel approach for constructing affine lower bounds for rational functions using Bernstein control points and convex hulls, with proven convergence properties.

Findings

Convergence of bounds improves with increased degree and subdivision.

The new method outperforms traditional Bernstein constant bounds in numerical tests.

Provides a systematic way to obtain tight linear bounds for polynomials and rational functions.

Abstract

In this paper, we investigate the problem of finding tight linear lower bounding functions for multivariate polynomials over boxes. These functions are obtained by the expansion of polynomials into Bernstein form and using the linear least squares function. Convergence properties of the given polynomials to their lower bounds are shown with respect to raising the degree, width of the box and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate rational function based on the Bernstein control points, the convex hull of a non-positive polynomial $s$ and degree elevation. Numerical comparisons with the well known Bernstein constant lower bounding function are finally given.