• 1. Introduction
  • The bernstein expansion and its applications




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    THE BERNSTEIN EXPANSION AND ITS APPLICATIONS
    Jürgen Garloff

    University of Applied Sciences/ FH Konstanz

    Department of Computer Science

    Postfach 100543

    D-78405 Konstanz

    Germany


    email: garloff@fh-konstanz.de

    Abstract: This tutorial article gives an introduction into the expansion of a multivariate polynomial into Bernstein polynomials and the use of this expansion for finding tight bounds for the range of the polynomial over a given box. Applications to robust stability problems, to the enclosure of the solu­tion set of systems of polynomial inequalities and equations, respectively, as well as to the solution of constrained global optimization problems are presented.

    Keywords: Bernstein polynomials, range enclosure, robust stability, polynomial inequalities, poly­nomial equations, constrained global optimization


    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.




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