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Institutional Repository - Research Portal Dépôt Institutionnel - Portail De La Recherche Institutional Repository - Research Portal Dépôt Institutionnel - Portail de la Recherche University of Namurresearchportal.unamur.be THESIS / THÈSE DOCTOR OF SCIENCES Filter-trust-region methods for nonlinear optimization Author(s) - Auteur(s) : Sainvitu, Caroline Award date: 2007 Awarding institution: University of Namur Supervisor - Co-Supervisor / Promoteur - Co-Promoteur : Link to publication Publication date - Date de publication : Permanent link - Permalien : Rights / License - Licence de droit d’auteur : General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. BibliothèqueDownload date: Universitaire 23. Jun. 2020 Moretus Plantin FACULTES UNIVERSITAIRES NOTRE-DAME DE LA PAIX NAMUR FACULTE DES SCIENCES DEPARTEMENT DE MATHEMATIQUE Filter-Trust-Region Methods for Nonlinear Optimization Dissertation présentée par Caroline Sainvitu pour l'obtention du grade de Docteur en Sciences Composition du Jury: Nick GOULD Annick SARTENAER Jean-Jacques STRODIOT Philippe TOINT (Promoteur) Luís VICENTE 2007 c Presses universitaires de Namur & Caroline Sainvitu Rempart de la Vierge, 13 B-5000 Namur (Belgique) Toute reproduction d'un extrait quelconque de ce livre, hors des limites restrictives prévues par la loi, par quelque procédé que ce soit, et notamment par photocopie ou scanner, est strictement interdite pour tous pays. Imprimé en Belgique ISBN-13 : 978-2-87037-548-8 Dépôt légal: D / 2007 / 1881 / 11 Facultés Universitaires Notre-Dame de la Paix Faculté des Sciences rue de Bruxelles, 61, B-5000 Namur, Belgium d a Facultés Universitaires Notre-Dame de la Paix Faculté des Sciences Rue de Bruxelles, 61, B-5000 Namur, Belgium Méthodes de filtre et de région de confiance pour l'optimisation non-linéaire par Caroline Sainvitu Résumé: Ce travail a pour objet l'étude théorique et l'implémentation d'algorithmes per- mettant la résolution de deux types particuliers de problèmes d'optimisation non-linéaire, à savoir les problèmes d'optimisation sans contrainte et avec contraintes de bornes. Pour l'optimisation sans contrainte, nous développons un nouvel algorithme qui utilise une tech- nique de filtre et une méthode de type région de confiance dans le but de garantir une conver- gence globale et d'améliorer l'efficacité des approches traditionnelles. Nous analysons égale- ment l'effet de dérivées premières et secondes approximées sur la performance de l'algorithme de filtre et de région de confiance. Nous étendons ensuite notre algorithme aux problèmes d'optimisation avec contraintes de bornes en combinant ces idées avec une méthode de pro- jection du gradient. Des résultats numériques accompagnent les méthodes proposées et in- diquent qu'elles sont compétitives par rapport aux méthodes de région de confiance plus classiques. Filter-trust-region methods for nonlinear optimization by Caroline Sainvitu Abstract: This work is concerned with the theoretical study and the implementation of al- gorithms for solving two particular types of nonlinear optimization problems, namely uncon- strained and simple-bound constrained optimization problems. For unconstrained optimiza- tion, we develop a new algorithm which uses a filter technique and a trust-region method in order to enforce global convergence and to improve the efficiency of traditional approaches. We also analyze the effect of approximate first and second derivatives on the performance of the filter-trust-region algorithm. We next extend our algorithm to simple-bound constrained optimization problems by combining these ideas with a gradient-projection method. Numer- ical results follow the proposed methods and indicate that they are competitive with more classical trust-region algorithms. Dissertation doctorale en Sciences mathématiques (Ph.D. thesis in Mathematics) Date: 17-04-2007 Département de Mathématique Promoteur (Advisor): Prof. Ph. L. TOINT c b Remerciements Je tiens tout d'abord à exprimer toute ma reconnaissance à Philippe Toint, mon promoteur, pour m'avoir acceuillie au sein de l'Unité d'Analyse Numérique ainsi que pour son encadrement, ses nombreux conseils et son soutien constant tout au long de cette thèse. Je le remercie également de m'avoir donné l'opportunité de participer à plusieurs conférences internationales. Ce fut pour moi une chance d'y présenter à chaque fois notre travail et d'y rencontrer de nombreux chercheurs. Je tiens également à remercier Nick Gould pour sa collaboration à une partie de ce travail et pour avoir accepté d'être dans le jury de cette thèse. Merci aussi à Annick Sartenaer, Jean-Jacques Strodiot et Luís Vicente d'avoir accepté de faire partie du jury de cette thèse. Je tiens aussi à remercier Dominique Orban de m'avoir invitée à plusieurs reprises à présenter mes recherches lors de conférences ainsi que tous les chercheurs que j'ai pu y rencontrer pour les discussions intéressantes ainsi que les moments de détente. Je remercie notamment Andreas Wächter pour ses suggestions pertinentes. Merci tout particulier à Katia Demaseure pour son amitié, sa bonne humeur et sa disponibilité. Je la remercie vivement pour toutes ces années passées ensemble au bout du couloir. Merci aussi à mes anciens collègues d'Analyse Numérique, à savoir Benoît Colson, qui fut toujours présent dans les moments de doutes, et Fabian Bastin, dont la distraction légendaire nous a valu beaucoup de fous rires. Mes plus chaleureux remerciements s'adressent également à tous les copains du midi avec qui j'ai eu la chance de partager pas mal de repas, pauses café et sorties. Merci à Charlotte Beau- thier, Florent Deleflie, Anne-Sophie Libert, Dimitri Tomanos, Stéphane Valk, Emilie Wanufelle, Melissa Weber Mendonça, Sebastian Xhonneux et tous les autres. Mes remerciements vont aussi vers tous les membres du département de Mathématique pour leur accueil et leur convivialité durant ces cinq années. Merci notamment à Eric Cornélis, i ii Remerciements Murielle Haguinet, Pascale Hermans et Martine Van Caenegem. Mes remerciements s'adressent également à Jean-Claude Lion, mon professeur de Mathéma- tique de secondaire, qui est sans nul doute à l'origine de mon goût pour les mathématiques. Merci aussi à tous mes amis, mathématiciens ou non, qui m'ont aidée, parfois à leur insu, par tous ces moments de détente passés en leur compagnie. Je tiens également à remercier mes parents qui ont toujours été présents dans les moments dif- ficiles et sans qui je ne serais pas où j'en suis aujourd'hui. Enfin, pour tout le reste et bien plus encore, je remercie Olivier Jacquet. Merci d'avoir été à mes côtés durant la longue et difficile période de la rédaction de cette thèse. Merci de m'avoir écoutée patiemment, d'avoir supporté les sautes d'humeur très fréquentes ces derniers mois et les remises en cause. A tous, encore merci. Caroline Contents Introduction vii 1 Generalities on optimization 1 1.1 What is optimization? . 1 1.2 Mathematical programming . 2 1.2.1 Formulation . 2 1.2.2 Classification of mathematical programs . 3 1.3 Basic notions . 6 1.3.1 Mathematical background . 6 1.3.2 Generalities on convergence . 9 1.3.3 Derivatives . 11 1.3.4 Approximation to derivatives . 13 1.3.5 Automatic differentiation . 15 2 Background on nonlinear optimization 17 2.1 Characterization of solutions . 17 2.2 Optimality conditions . 18 2.2.1 Unconstrained optimization . 19 2.2.2 Constrained optimization . 20 2.2.3 Criticality measure . 24 2.3 Methods for nonlinear unconstrained optimization . 24 2.3.1 Local methods . 24 2.3.2 Line-search methods . 29 2.3.3 Trust-region methods . 29 2.3.4 Conjugate-gradient methods . 32 2.4 Methods for nonlinear constrained optimization . 33 2.4.1 Penalty methods . 34 2.4.2 Augmented Lagrangian methods . 36 iii iv CONTENTS 2.4.3 Sequential Quadratic Programming . 38 2.5 References . 40 3 A quick survey of filter methods 41 3.1 Motivation of the filter . 41 3.2 The first filter approach . 42 3.3 Bibliographical review . 47 I Unconstrained Optimization 49 4 A filter-trust-region method for unconstrained optimization 51 4.1 The problem and the new algorithm . 52 4.1.1 Computing a trial point . 53 4.1.2 The multidimensional filter . 54 4.1.3 The filter-trust-region algorithm . 56 4.2 Convergence analysis . 58 4.2.1 Assumptions and notations . 58 4.2.2 Convergence to first-order critical points . 60 4.2.3 Analysis of a counter-example . 66 4.2.4 Convergence to second-order critical points . 70 4.3 Conclusion . 72 5 Numerical results 73 5.1 Testing environment . 73 5.2 Performance profiles . 76 5.3 Practical aspects . 77 5.4 Performance and comparisons . 80 5.4.1 Filter versus pure trust-region variants . 80 5.4.2 Comparison on quadratic programs . 84 5.4.3 Comparison with LANCELOT-B . 87 5.4.4 Algorithmic variants . 93 5.4.5 Unrestricted steps . 97 5.5 Conclusion . 99 6 Do approximate derivatives hurt filter methods? 101 6.1 The framework . 101 6.2 Numerical investigation . 103 CONTENTS v 6.2.1 Finite differences . 105 6.2.2 Secant updates . 113 6.2.3 Perturbation of the Hessian . 121 6.2.4 Comparison . 121 6.3 Conclusion . 125 II Bound-Constrained Optimization 127 7 A filter-trust-region method for bound-constrained optimization 129 7.1 Introduction to bound-constrained optimization .
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