Minimization and Positivity of the Tensorial Rational Bernstein Form

Abstract

Rational functions of total degree l in n variables have a representation in the Bernstein form defined over n dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein coefficients over a simplex. Convergence properties of the bounds to the range are reviewed. Algebraic identities certifying the positivity of a given rational function over a simplex are given. Subsequently, a bound established in this work does not depend on the given dimension.

Direct Algorithm for Bernstein Enclosure Boundary of Polynomials

“…In [9], the authors published results in degree elevation and subdivision of the underlying simplex of Bernstein basis for solving global optimization and system problems. In [10,11], the tensorial Bernstein case over boxes was addressed for computing the enclosure range of a given (multivariate) rational polynomial function, which is slow. Generally speaking, minimizing and maximizing of Bernstein coe cients provide bounds for the range of its polynomial function F over any given simplex, whereas the complexity of computing these coe cients is high.…”

Linear Optimization of Polynomial Rational Functions: Applications for Positivity Analysis

“…Bernstein polynomials and their bounding functions were used in a huge variety of problems; see [4,5]. Different methods of minimization and deriving a lower bound have been attacked previously by the expansion of a given polynomial into Bernstein form [1,[6][7][8][9][10]; until now, the best lower bound is a linear function relying on the control points and the convex hull property; which is illustrated in Figure 1. Affine lower bounding functions for polynomials on a box were studied in [9], however without any proof of the convergence for the error bound between the affine lower bound and its (monomial) polynomial.…”