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

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

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

@unpublished{Garloff2008Rigor-629, title={Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation}, year={2008},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

<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bibo="http://purl.org/ontology/bibo/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > <rdf:Description rdf:about="https://kops.uni-konstanz.de/rdf/resource/123456789/629"> <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-03-22T17:45:17Z</dcterms:available> <dcterms:title>Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation</dcterms:title> <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2011-03-22T17:45:17Z</dc:date> <dcterms:abstract xml:lang="eng">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.</dcterms:abstract> <dc:contributor>Smith, Andrew Paul</dc:contributor> <dc:creator>Garloff, Jürgen</dc:creator> <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/629"/> <dcterms:rights rdf:resource="http://nbn-resolving.org/urn:nbn:de:bsz:352-20140905103416863-3868037-7"/> <dc:language>eng</dc:language> <dcterms:issued>2008</dcterms:issued> <dc:contributor>Garloff, Jürgen</dc:contributor> <dc:format>application/pdf</dc:format> <dc:rights>deposit-license</dc:rights> <dc:creator>Smith, Andrew Paul</dc:creator> </rdf:Description> </rdf:RDF>

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