A Theoretical and Numerical Study of a New Hybrid Conjugate Gradient Method for Nonlinear Programming / Lamya Mendil

Public ISBD Titre : A Theoretical and Numerical Study of a New Hybrid Conjugate Gradient Method for Nonlinear Programming Type de document : texte imprimé Auteurs : Lamya Mendil, Auteur ; Chalabia Tebbal ; Assma Leulmi, Directeur de thèse Editeur : Sétif:UFS Année de publication : 2024 Importance : 1 vol (56 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Unconstrained nonlinear optimization Conjugate gradient method Hybrid method Inexact line search The convergence Index. décimale : 510-Mathématique Résumé : The conjugate gradient method is one of the oldest methods for solving nonlinear unconstrained optimization problems especially in large size. This memory present a new hybrid conjugate gradient method based on the convex combination of NHS (modified HS) and MLS (modified LS) methods. The proposed algorithm satisfies the sufficient descent condition and converges globally under the usual and strong Wolfe line search assumptions. To illustrate the effectiveness of this method a numerical study is achieved. Note de contenu : Sommaire Introductioniii 1 Preliminariesandbasicconcepts1 1.1GeneraldeÂ…nitionsonunconstrainedoptimization..........1 1.2Existenceanduniquenessresults....................4 1.3Conditionsofoptimality........................5 1.3.1Necessaryconditions(NC)...................5 1.3.2Su¢cientConditions(SC)...................6 1.4Descentdirectionmethod.......................6 1.4.1Iterativealgorithm.......................7 2 Exactandinexactlinesearch9 2.1Purposeoflinesearch.........................9 2.2Safetyinterval..............................10 2.3Basicalgorithm.............................10 2.4Exactlinesearchmethods.......................10 2.5Inexactlinesearchmethods......................11 2.5.1Armijorule(1966).......................12 2.5.2Goldstein-Pricerule(1969)...................13 2.5.3Wolferule(1969)........................14 3 Theconjugategradientmethod18 3.1Gradientmethod............................18 3.1.1Gradientmethodalgorithm..................19 3.1.2convergenceofthegradientmethod..............19 3.2Theconjugategradientmethodfornonlinearfunctions.......20 3.2.1Generalprinciple........................20 3.2.2Somevariantsofthenonlinearconjugategradientmethod.21 3.2.3NonlinearconjugategradientalgorithmwithstrongWolfeÂ’s inexactlinesearch.......................21 3.3Convergenceresultsfortheconjugategradientmethod.......22 3.3.1Conditions C1 and C2 (Lipschitzandboundness)......22 3.3.2ZoutendijkÂ’sTheorem.....................23 3.3.3UsingZoutendijkÂ’stheoremtodemonstrateglobalconvergence23 4 Hybridmethodofnonlinearconjugategradient27 4.1HybridmethodsbasedonthemodiÂ…cationofclassicalconjugate gradientmethods............................27 4.1.1ThemodiÂ…cationofPRPmethod...............27 4.1.2ThemodiÂ…cationofFRmethod................28 4.1.3ThemodiÂ…cationofHSmethod................28 4.1.4ThemodiÂ…cationofLSmethod................29 4.2Hybridnonlinearconjugategradientmethodsbasedonconvexcom- binations.................................29 4.2.1Generalprinciple........................29 4.3AnewhybridCGmethodbasedonconvexcombination......30 4.3.1Theproposedalgorithm....................30 4.3.2Thesu¢cientdescentconditionandtheglobalconvergence.31 5 Numericalapplications 35 Côte titre : MAM/0727 A Theoretical and Numerical Study of a New Hybrid Conjugate Gradient Method for Nonlinear Programming [texte imprimé] / Lamya Mendil, Auteur ; Chalabia Tebbal ; Assma Leulmi, Directeur de thèse . - [S.l.] : Sétif:UFS, 2024 . - 1 vol (56 f.) ; 29 cm. Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Unconstrained nonlinear optimization Conjugate gradient method Hybrid method Inexact line search The convergence Index. décimale : 510-Mathématique Résumé : The conjugate gradient method is one of the oldest methods for solving nonlinear unconstrained optimization problems especially in large size. This memory present a new hybrid conjugate gradient method based on the convex combination of NHS (modified HS) and MLS (modified LS) methods. The proposed algorithm satisfies the sufficient descent condition and converges globally under the usual and strong Wolfe line search assumptions. To illustrate the effectiveness of this method a numerical study is achieved. Note de contenu : Sommaire Introductioniii 1 Preliminariesandbasicconcepts1 1.1GeneraldeÂ…nitionsonunconstrainedoptimization..........1 1.2Existenceanduniquenessresults....................4 1.3Conditionsofoptimality........................5 1.3.1Necessaryconditions(NC)...................5 1.3.2Su¢cientConditions(SC)...................6 1.4Descentdirectionmethod.......................6 1.4.1Iterativealgorithm.......................7 2 Exactandinexactlinesearch9 2.1Purposeoflinesearch.........................9 2.2Safetyinterval..............................10 2.3Basicalgorithm.............................10 2.4Exactlinesearchmethods.......................10 2.5Inexactlinesearchmethods......................11 2.5.1Armijorule(1966).......................12 2.5.2Goldstein-Pricerule(1969)...................13 2.5.3Wolferule(1969)........................14 3 Theconjugategradientmethod18 3.1Gradientmethod............................18 3.1.1Gradientmethodalgorithm..................19 3.1.2convergenceofthegradientmethod..............19 3.2Theconjugategradientmethodfornonlinearfunctions.......20 3.2.1Generalprinciple........................20 3.2.2Somevariantsofthenonlinearconjugategradientmethod.21 3.2.3NonlinearconjugategradientalgorithmwithstrongWolfeÂ’s inexactlinesearch.......................21 3.3Convergenceresultsfortheconjugategradientmethod.......22 3.3.1Conditions C1 and C2 (Lipschitzandboundness)......22 3.3.2ZoutendijkÂ’sTheorem.....................23 3.3.3UsingZoutendijkÂ’stheoremtodemonstrateglobalconvergence23 4 Hybridmethodofnonlinearconjugategradient27 4.1HybridmethodsbasedonthemodiÂ…cationofclassicalconjugate gradientmethods............................27 4.1.1ThemodiÂ…cationofPRPmethod...............27 4.1.2ThemodiÂ…cationofFRmethod................28 4.1.3ThemodiÂ…cationofHSmethod................28 4.1.4ThemodiÂ…cationofLSmethod................29 4.2Hybridnonlinearconjugategradientmethodsbasedonconvexcom- binations.................................29 4.2.1Generalprinciple........................29 4.3AnewhybridCGmethodbasedonconvexcombination......30 4.3.1Theproposedalgorithm....................30 4.3.2Thesu¢cientdescentconditionandtheglobalconvergence.31 5 Numericalapplications 35 Côte titre : MAM/0727

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