Publikation:

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

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2008

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Smith, Andrew Paul

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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.

Zusammenfassung in einer weiteren Sprache

Fachgebiet (DDC)
510 Mathematik

Schlagwörter

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

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ISO 690GARLOFF, 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@unpublished{Garloff2008Rigor-629,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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