1. Introduction
Many problems in pure and applied mathematics and their applications can be reduced to the problem of finding bounds for the range of a given function over a prescribed domain. Often this
domain is an axisparallel box X in
Rn. E.g., in robust control and computer aided geometric design it is often required to check whether the determinant of a matrix with entries depending on parameters varying in intervals is of like sign for all parameter combinations. If the given function is a multivariate polynomial,
p say,
the expansion of p into Bernstein polynomials provides tight bounds on its range. The aim of this tutorial article is to give an introduction into this expansion and to present various applications of these bounds.
The organization of this article is as follows: In the next section we recall the Bernstein expansion. In Section 3 applications to some robust control problems are presented. Bounding the solution sets of systems of polynomial inequalities and equations are treated in Sections 4 and 5, respectively. Applications to the solution of constrained global optimization problems are given in Section 6. Directions for further research are outlined in the last section. To keep the presentation as simple as possible, we focus on the applications and refer to papers, where the underlying theory can be found.