Publikation: Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation
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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.
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GARLOFF, Jürgen, Andrew Paul SMITH, 2008. Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global OptimisationBibTex
@unpublished{Garloff2008Rigor-629, year={2008}, title={Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation},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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