A brief review of a method for bounds on polynomial ranges over simplexes

Abstract

We are concerned with tools to find bounds on the range of certain polynomial functions of n variables. Although our motivation and history of the tools are from crisp global optimization, bounding the range of such functions is also important in fuzzy logic implementations. We review and provide a new perspective on one such tool. We have been examining problems naturally posed in terms of barycentric coordinates, that is, over simplexes. There is a long history of using Bernstein expansions to bound ranges of polynomials over simplexes, particularly within the computer graphics community for 1-, 2-, and 3-dimensional problems, with some literature on higher-dimensional generalizations, and some work on use in global optimization. We revisit this work, identifying efficient implementation and practical application contexts, to bound ranges of polynomials over simplexes in dimensions, 2, 3, and higher.