Numerical optimization and positivity certificates for polynomials and rationals over simplices

Multivariate rational control functions r(x) of high degree d can be expressed in the Bernstein form defined over k-dimensional simplices (triangles). The range of rational functions can be optimized by the enclosure bound (minimum and maximum) of Bernstein coefficients. In this paper, linear and quadratic rate of convergence for the maximum error bound between the enclosure and range bounds are investigated. Furthermore, minimization and numerical identities for certifying the positivity of any given rational function over simplices are given. In order to provide global and local positivity certificates for r, the degree of Bernstein functions and number of simplex subdivision steps are estimated. Subsequently, we establish a bound for the maximum degree of Bernstein polynomials.