Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation

This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. The coefficients of this expansion over a given box yield a control point structure whose convex hull contains the graph of the given polynomial over the box. We introduce a new method for computing tight affine lower bound functions based on these control points, using a linear least squares approximation of the entire control point structure. This is demonstrated to have superior performance to previous methods based on a linear interpolation of certain specially chosen control points. The problem of how to obtain a verified affine lower bound function in the presence of uncertainty and rounding errors is also considered. Numerical results with error bounds for a series of randomly-generated polynomials are given.