Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation

Garloff, Jürgen; Smith, Andrew Paul

Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation

Dateien

Rigorous_Affine_Lower_Bound_Functions_for_Multivariate_Polynomials_and_Their_Use_in_Global_Optimisation.pdfGröße: 357.04 KBDownloads: 310

Datum

2008

Autor:innen

Garloff, Jürgen

Smith, Andrew Paul

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Mathematik und Statistik

Publikationstyp

Preprint

Publikationsstatus

Published

Zusammenfassung

This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. The coefficients of this expansion over a given box yield a control point structute whose convex hull contains the graph of the given polynomial over the box. We introduce a new method for computing tight affine lower bound functions based on these control points, using a linear least squares approximation of the entire control point structure. This is demonstrated to have superior performance to previous methods based on a linear interpolation of certain specially chosen control points. The problem of how to obtain a verfied affine lower bound function in the presence of uncertainty and rounding errors is also considered. Numerical results with error bounds for a series of randomly-generated polynomials are given.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Bernstein-Polynom, global optimisation, branch and bound method, multivariate polynomial, Bernstein polynomial

Zitieren

ISO 690

GARLOFF, Jürgen, Andrew Paul SMITH, 2008. Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation

BibTex

@unpublished{Garloff2008Rigor-629,
  year={2008},
  title={Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation},
  author={Garloff, Jürgen and Smith, Andrew Paul}
}

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