Constraint augmentation in pseudo-singularly perturbed linear programs
Identifieur interne : 005D61 ( Main/Exploration ); précédent : 005D60; suivant : 005D62Constraint augmentation in pseudo-singularly perturbed linear programs
Auteurs : K. Avrachenkov [France] ; R. S. Burachik [Australie] ; J. A. Filar [Australie] ; V. Gaitsgory [Australie]Source :
- Mathematical programming [ 0025-5610 ] ; 2012.
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Abstract
In this paper we study a linear programming problem with a linear perturbation introduced through a parameter ε > 0. We identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. We show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.
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Avrachenkov</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>INRIA. 2004 route des lucioles-BP 93</s1><s2>06902 Sophia Antipolis</s2><s3>FRA</s3><sZ>1 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region" nuts="2">Provence-Alpes-Côte d'Azur</region><settlement type="city">Sophia Antipolis</settlement></placeName></affiliation></author><author><name sortKey="Burachik, R S" sort="Burachik, R S" uniqKey="Burachik R" first="R. S." last="Burachik">R. S. Burachik</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Centre for Industrial and Applied Mathematics, School of Mathematics and Statistics, University of South Australia</s1><s2>Mawson Lakes, SA 5095</s2><s3>AUS</s3><sZ>2 aut.</sZ><sZ>3 aut.</sZ><sZ>4 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Mawson Lakes, SA 5095</wicri:noRegion></affiliation></author><author><name sortKey="Filar, J A" sort="Filar, J A" uniqKey="Filar J" first="J. A." last="Filar">J. A. Filar</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Centre for Industrial and Applied Mathematics, School of Mathematics and Statistics, University of South Australia</s1><s2>Mawson Lakes, SA 5095</s2><s3>AUS</s3><sZ>2 aut.</sZ><sZ>3 aut.</sZ><sZ>4 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Mawson Lakes, SA 5095</wicri:noRegion></affiliation></author><author><name sortKey="Gaitsgory, V" sort="Gaitsgory, V" uniqKey="Gaitsgory V" first="V." last="Gaitsgory">V. Gaitsgory</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Centre for Industrial and Applied Mathematics, School of Mathematics and Statistics, University of South Australia</s1><s2>Mawson Lakes, SA 5095</s2><s3>AUS</s3><sZ>2 aut.</sZ><sZ>3 aut.</sZ><sZ>4 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Mawson Lakes, SA 5095</wicri:noRegion></affiliation></author></analytic><series><title level="j" type="main">Mathematical programming</title><title level="j" type="abbreviated">Math. program.</title><idno type="ISSN">0025-5610</idno><imprint><date when="2012">2012</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Mathematical programming</title><title level="j" type="abbreviated">Math. program.</title><idno type="ISSN">0025-5610</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Asymptotic approximation</term><term>Constrained optimization</term><term>Constraint</term><term>Discontinuity</term><term>Iterative method</term><term>Lagrange multiplier</term><term>Linear programming</term><term>Modeling</term><term>Objective function</term><term>Perturbation method</term><term>Sensitivity analysis</term><term>Singular perturbation</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Contrainte</term><term>Perturbation singulière</term><term>Méthode perturbation</term><term>Programmation linéaire</term><term>Approximation asymptotique</term><term>Discontinuité</term><term>Fonction objectif</term><term>Optimisation sous contrainte</term><term>Multiplicateur Lagrange</term><term>Méthode itérative</term><term>Analyse sensibilité</term><term>Modélisation</term><term>.</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">In this paper we study a linear programming problem with a linear perturbation introduced through a parameter ε > 0. We identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. We show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.</div></front></TEI><affiliations><list><country><li>Australie</li><li>France</li></country><region><li>Provence-Alpes-Côte d'Azur</li></region><settlement><li>Sophia Antipolis</li></settlement></list><tree><country name="France"><region name="Provence-Alpes-Côte d'Azur"><name sortKey="Avrachenkov, K" sort="Avrachenkov, K" uniqKey="Avrachenkov K" first="K." last="Avrachenkov">K. Avrachenkov</name></region></country><country name="Australie"><noRegion><name sortKey="Burachik, R S" sort="Burachik, R S" uniqKey="Burachik R" first="R. S." last="Burachik">R. S. Burachik</name></noRegion><name sortKey="Filar, J A" sort="Filar, J A" uniqKey="Filar J" first="J. A." last="Filar">J. A. Filar</name><name sortKey="Gaitsgory, V" sort="Gaitsgory, V" uniqKey="Gaitsgory V" first="V." last="Gaitsgory">V. Gaitsgory</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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