Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems
Identifieur interne : 00BB84 ( Main/Exploration ); précédent : 00BB83; suivant : 00BB85Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems
Auteurs : A. Fischer [Allemagne] ; V. Jeyakumar [Australie] ; D. T. Luc [France]Source :
- Journal of optimization theory and applications [ 0022-3239 ] ; 2001.
Descripteurs français
- Pascal (Inist)
- Problème non linéaire, Problème complémentarité, Continu, Analyse non lisse, Fonction jacobienne, Méthode descente, Non linéarité, Condition stationnaire, Algorithme, Condition nécessaire, Condition optimalité, Convergence, Solution, Méthode approchée, Critère optimalité, 0230Y, Fischer-Burmeister merit fuction, Approximate Jacobians, Nonsmooth continuous maps, Stationary points, Global convergence, Derivative-free algorithm, Directional monotonicity, Convergence globale.
English descriptors
- KwdEn :
Abstract
We consider a nonlinear complementarity problem defined by a continuous but not necessarily locally Lipschitzian map. In particular, we examine the connection between solutions of the problem and stationary points of the associated Fischer-Burmeister merit function. This is done by deriving a new necessary optimality condition and a chain rule formula for composite functions involving continuous maps. These results are given in terms of approximate Jacobians which provide the foundation for analyzing continuous nonsmooth maps. We also prove a result on the global convergence of a derivative-free descent algorithm for solving the complementarity problem. To this end, a concept of directional monotonicity for continuous maps is introduced.
Affiliations:
- Allemagne, Australie, France
- District d'Arnsberg, Provence-Alpes-Côte d'Azur, Rhénanie-du-Nord-Westphalie
- Avignon, Dortmund
- Université d'Avignon
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Le document en format XML
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Jeyakumar</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Department of Applied Mathematics, University of New South Wales</s1><s2>Sydney, NSW</s2><s3>AUS</s3><sZ>2 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Sydney, NSW</wicri:noRegion></affiliation></author><author><name sortKey="Luc, D T" sort="Luc, D T" uniqKey="Luc D" first="D. T." last="Luc">D. T. 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Fischer</name><affiliation wicri:level="3"><inist:fA14 i1="01"><s1>Department of Mathematics, University of Dortmund</s1><s2>Dortmund</s2><s3>DEU</s3><sZ>1 aut.</sZ></inist:fA14><country>Allemagne</country><placeName><region type="land" nuts="1">Rhénanie-du-Nord-Westphalie</region><region type="district" nuts="2">District d'Arnsberg</region><settlement type="city">Dortmund</settlement></placeName></affiliation></author><author><name sortKey="Jeyakumar, V" sort="Jeyakumar, V" uniqKey="Jeyakumar V" first="V." last="Jeyakumar">V. Jeyakumar</name><affiliation wicri:level="1"><inist:fA14 i1="02"><s1>Department of Applied Mathematics, University of New South Wales</s1><s2>Sydney, NSW</s2><s3>AUS</s3><sZ>2 aut.</sZ></inist:fA14><country>Australie</country><wicri:noRegion>Sydney, NSW</wicri:noRegion></affiliation></author><author><name sortKey="Luc, D T" sort="Luc, D T" uniqKey="Luc D" first="D. T." last="Luc">D. T. Luc</name><affiliation wicri:level="4"><inist:fA14 i1="03"><s1>Departement de Mathématiques, Université d'Avignon</s1><s2>Avignon</s2><s3>FRA</s3><sZ>3 aut.</sZ></inist:fA14><country>France</country><placeName><region type="region">Provence-Alpes-Côte d'Azur</region><region type="old region">Provence-Alpes-Côte d'Azur</region><settlement type="city">Avignon</settlement><settlement type="city">Avignon</settlement></placeName><orgName type="university">Université d'Avignon</orgName></affiliation></author></analytic><series><title level="j" type="main">Journal of optimization theory and applications</title><title level="j" type="abbreviated">J. optim. theory appl.</title><idno type="ISSN">0022-3239</idno><imprint><date when="2001">2001</date></imprint></series></biblStruct></sourceDesc><seriesStmt><title level="j" type="main">Journal of optimization theory and applications</title><title level="j" type="abbreviated">J. optim. theory appl.</title><idno type="ISSN">0022-3239</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Algorithms</term><term>Approximate method</term><term>Complementarity problem</term><term>Continuous</term><term>Convergence</term><term>Descent method</term><term>Global convergence</term><term>Jacobian function</term><term>Necessary condition</term><term>Nonlinear problems</term><term>Nonlinearity</term><term>Nonsmooth analysis</term><term>Optimality condition</term><term>Optimality criterion</term><term>Solutions</term><term>Stationary condition</term></keywords><keywords scheme="Pascal" xml:lang="fr"><term>Problème non linéaire</term><term>Problème complémentarité</term><term>Continu</term><term>Analyse non lisse</term><term>Fonction jacobienne</term><term>Méthode descente</term><term>Non linéarité</term><term>Condition stationnaire</term><term>Algorithme</term><term>Condition nécessaire</term><term>Condition optimalité</term><term>Convergence</term><term>Solution</term><term>Méthode approchée</term><term>Critère optimalité</term><term>0230Y</term><term>Fischer-Burmeister merit fuction</term><term>Approximate Jacobians</term><term>Nonsmooth continuous maps</term><term>Stationary points</term><term>Global convergence</term><term>Derivative-free algorithm</term><term>Directional monotonicity</term><term>Convergence globale</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">We consider a nonlinear complementarity problem defined by a continuous but not necessarily locally Lipschitzian map. In particular, we examine the connection between solutions of the problem and stationary points of the associated Fischer-Burmeister merit function. This is done by deriving a new necessary optimality condition and a chain rule formula for composite functions involving continuous maps. These results are given in terms of approximate Jacobians which provide the foundation for analyzing continuous nonsmooth maps. We also prove a result on the global convergence of a derivative-free descent algorithm for solving the complementarity problem. To this end, a concept of directional monotonicity for continuous maps is introduced.</div></front></TEI><affiliations><list><country><li>Allemagne</li><li>Australie</li><li>France</li></country><region><li>District d'Arnsberg</li><li>Provence-Alpes-Côte d'Azur</li><li>Rhénanie-du-Nord-Westphalie</li></region><settlement><li>Avignon</li><li>Dortmund</li></settlement><orgName><li>Université d'Avignon</li></orgName></list><tree><country name="Allemagne"><region name="Rhénanie-du-Nord-Westphalie"><name sortKey="Fischer, A" sort="Fischer, A" uniqKey="Fischer A" first="A." last="Fischer">A. Fischer</name></region></country><country name="Australie"><noRegion><name sortKey="Jeyakumar, V" sort="Jeyakumar, V" uniqKey="Jeyakumar V" first="V." last="Jeyakumar">V. Jeyakumar</name></noRegion></country><country name="France"><region name="Provence-Alpes-Côte d'Azur"><name sortKey="Luc, D T" sort="Luc, D T" uniqKey="Luc D" first="D. T." last="Luc">D. T. 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