Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

In this paper, we provide tight linear lower bounding functions for multivariate polynomials given over boxes. These functions are obtained by the expansion of polynomials into Bernstein basis and using the linear least squares function. Convergence properties for the absolute difference between the given polynomials and their lower bounds are shown with respect to raising the degree and the width of boxes and subdivision. Subsequently, we provide a new method for constructing an affine lower bounding function for a multivariate continuous 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 given. Finally, with these affine functions, the positivity of polynomials and rational functions can be certified by computing the Bernstein coefficients of their linear lower bounds.