New LP bound in multivariate Lipschitz optimization: Theory and applications
Identifieur interne : 002946 ( Main/Exploration ); précédent : 002945; suivant : 002947New LP bound in multivariate Lipschitz optimization: Theory and applications
Auteurs : R. Horst [Allemagne] ; M. Nast [Allemagne] ; N. V. Thoai [Viêt Nam]Source :
- Journal of Optimization Theory and Applications [ 0022-3239 ] ; 1995-08-01.
Descripteurs français
- Pascal (Inist)
English descriptors
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Abstract
Abstract: Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.
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DOI: 10.1007/BF02192085
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Thoai</name><affiliation wicri:level="1"><country xml:lang="fr">Viêt Nam</country><wicri:regionArea>Institute of Mathematics, Bo Ho, Hanoi</wicri:regionArea><wicri:noRegion>Hanoi</wicri:noRegion></affiliation></author></analytic><monogr></monogr><series><title level="j">Journal of Optimization Theory and Applications</title><title level="j" type="abbrev">J Optim Theory Appl</title><idno type="ISSN">0022-3239</idno><idno type="eISSN">1573-2878</idno><imprint><publisher>Kluwer Academic Publishers-Plenum Publishers</publisher><pubPlace>New York</pubPlace><date type="published" when="1995-08-01">1995-08-01</date><biblScope unit="volume">86</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="369">369</biblScope><biblScope unit="page" to="388">388</biblScope></imprint><idno type="ISSN">0022-3239</idno></series><idno type="istex">1A01A941FB2BB53038BCE09025E7E7CA3A7FAC81</idno><idno type="DOI">10.1007/BF02192085</idno><idno type="ArticleID">BF02192085</idno><idno type="ArticleID">Art5</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0022-3239</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Algorithm</term><term>Branch and bound method</term><term>Constrained optimization</term><term>Inequality</term><term>Iterative method</term><term>Linear programming</term><term>Lower bound</term><term>Method study</term><term>Numerical simulation</term><term>Problem solving</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Algorithme</term><term>Borne inférieure</term><term>Etude méthode</term><term>Inégalité</term><term>Méthode branch and bound</term><term>Méthode itérative</term><term>Optimisation Lipschitz</term><term>Optimisation sous contrainte</term><term>Programmation linéaire</term><term>Résolution problème</term><term>Simulation numérique</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. 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