Convergent bounds for the range of multivariate polynomials

Scientific Computing, Computer Arithmetic, and Validated Numerics

Hamadneh

2016

Abstract. A method is investigated by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials in Bernstein polynomials. Convergence of the bounds to the range with respect to degree elevation of the Bernstein expansion, to the width of the box and to subdivision are proven and the inclusion isotonicity of the related enclosure function is shown.

Section: The Polynomial Bernstein Formmentioning

confidence: 99%

Convergence and Inclusion Isotonicity of the Tensorial Rational Bernstein Form

Scientific Computing, Computer Arithmetic, and Validated Numerics

Hamadneh

2016

“…The latter ones require the ability to compute tight bounds for the range of the objective function and the functions describing the constraints over the considered search region. In the case of polynomial optimization problems, one can make use of the expansion of a polynomial into Bernstein polynomials, see [7], [16], [18], [19], [21], [22]. Then the minimum and maximum of the coefficients of this expansion, the so-called Bernstein coefficients, provide bounds for the range of the polynomial over the search region.…”

Section: Introductionmentioning

confidence: 99%

“…The advantages and disadvantages of the use of the simplicial Bernstein basis compared to the use of the tensorial Bernstein basis, e.g. [7], where the underlying region is a box of R n , are discussed in [23]. For a comprehensive survey on the properties of the Bernstein bases see [6].…”

Section: Introductionmentioning

confidence: 99%

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

Titi

Applied Mathematics and Computation

2017

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.

“…During the last decade, polynomial minimization over simplices has attracted the interest of many researchers, see [1,2], [4], [6][7][8][9][10][11][12]. Special attention was paid to the use of the expansion of the given polynomial into Bernstein polynomials over a simplex, the so-called simplicial Bernstein expansion, [2], [4], [8], [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…Special attention was paid to the use of the expansion of the given polynomial into Bernstein polynomials over a simplex, the so-called simplicial Bernstein expansion, [2], [4], [8], [10][11][12]. In [10][11][12], R. Leroy gave results on degree elevation and subdivision of the underlying simplex of this expansion.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the Simplicial Rational Bernstein Form

Titi

Hamadneh

Advances in Intelligent Systems and Computing

2015

Abstract. Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for the range of p over V . Bounds for the range of a rational function over V can easily obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds converge monotonically and linearly to the range of the rational function if the degree of the Bernstein expansion is elevated. If V is subdivided then the convergence is quadratic with respect to the maximum of the diameters of the subsimplices.