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An Active–Set Trust-Region Method for Bound-Constrained Nonlinear Optimization Without Derivatives Applied to Noisy Aerodynamic Design Problems Anke Troltzsch

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An Active–Set Trust-Region Method for Bound-Constrained Nonlinear Optimization Without Derivatives Applied to Noisy Aerodynamic Design Problems Anke Troltzsch An active–set trust-region method for bound-constrained nonlinear optimization without derivatives applied to noisy aerodynamic design problems Anke Troltzsch To cite this version: Anke Troltzsch. An active–set trust-region method for bound-constrained nonlinear optimization with- out derivatives applied to noisy aerodynamic design problems. Optimization and Control [math.OC]. Institut National Polytechnique de Toulouse - INPT, 2011. English. tel-00639257 HAL Id: tel-00639257 https://tel.archives-ouvertes.fr/tel-00639257 Submitted on 8 Nov 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ฀ ฀฀฀฀฀ ฀ %0$503"5%&%0$503"5%&-6/*7&34*5²-6/*7฀&34*5²%&506-064&%&506-064& ฀฀ Institut National Polytechnique฀ de Toulouse (INP Toulouse) ฀฀฀ Mathématiques Informatique฀ Télécommunications ฀ ฀฀฀฀฀฀ Anke TRÖLTZSCH฀ ฀ mardi 7 juin 2011 ฀ ฀ ฀ ฀ An active-set trust-region method for bound-constrained nonlinear optimization without฀ derivatives applied to noisy aerodynamic฀ design problems ฀ ฀฀฀ Mathématiques Informatique฀ Télécommunications (MITT) ฀฀฀฀ CERFACS ฀฀฀ Serge Gratton Philippe L. Toint ฀ Luis Nunes฀ Vicente Jean-Charles Gilbert ฀ ฀฀฀฀ Bijan Mohammadi Matthieu Meaux THÈSE présentée en vue de l’obtention du titre de DOCTEUR DE L’UNIVERSITÉ DE TOULOUSE Spécialité : Mathématiques, Informatique et Télécommunications par Anke TRÖLTZSCH CERFACS Une méthode de région de confiance avec ensemble actif pour l’optimisation non linéaire sans dérivées avec contraintes de bornes appliquée à des problèmes aérodynamiques bruités soutenue le 7 juin 2011 devant le jury composé de : Serge Gratton Directeur de thèse IRIT et CERFACS Philippe L. Toint Co-directeur de thèse Université de Namur Luis Nunes Vicente Rapporteur Université de Coimbra Jean-Charles Gilbert Rapporteur INRIA Paris - Rocquencourt Bijan Mohammadi Examinateur CERFACS Matthieu Meaux Examinateur Airbus France Réf. CERFACS : TH-PA-11-69 Résumé L’optimisation sans dérivées (OSD) a connu un regain d’intérêt ces dernières années, prin- cipalement motivée par le besoin croissant de résoudre les problèmes d’optimisation définis par des fonctions dont les valeurs sont calculées par simulation (par exemple, la conception technique, la restauration d’images médicales ou de nappes phréatiques). Ces dernières années, un certain nombre de méthodes d’optimisation sans dérivée ont été développées et en particulier des méthodes fondées sur un modèle de région de confiance se sont avérées obtenir de bons résultats. Dans cette thèse, nous présentons un nouvel algorithme de région de confiance, basé sur l’interpolation, qui se montre efficace et globalement convergent (en ce sens que sa convergence vers un point stationnaire est garantie depuis tout point de départ arbitraire). Le nouvel algo- rithme repose sur la technique d’auto-correction de la géométrie proposé par Scheinberg and Toint (2010). Dans leur théorie, ils ont fait avancer la compréhension du rôle de la géomé- trie dans les méthodes d’OSD à base de modèles. Dans notre travail, nous avons pu améliorer considérablement l’efficacité de leur méthode, tout en maintenant ses bonnes propriétés de convergence. De plus, nous examinons l’influence de différents types de modèles d’interpolation sur les performances du nouvel algorithme. Nous avons en outre étendu cette méthode pour prendre en compte les contraintes de borne par l’application d’une stratégie d’activation. Considérer une méthode avec ensemble actif pour l’optimisation basée sur des modèles d’interpolation donne la possibilité d’économiser une quan- tité importante d’évaluations de fonctions. Il permet de maintenir les ensembles d’interpolation plus petits tout en poursuivant l’optimisation dans des sous-espaces de dimension inférieure. L’algorithme résultant montre un comportement numérique très compétitif. Nous présentons des résultats sur un ensemble de problèmes-tests issu de la collection CUTEr et comparons notre méthode à des algorithmes de référence appartenant à différentes classes de méthodes d’OSD. Pour réaliser des expériences numériques qui intègrent le bruit, nous créons un ensemble de cas-tests bruités en ajoutant des perturbations à l’ensemble des problèmes sans bruit. Le choix des problèmes bruités a été guidé par le désir d’imiter les problèmes d’optimisation basés sur la simulation. Enfin, nous présentons des résultats sur une application réelle d’un problème de conception de forme d’une aile fourni par Airbus. Mots-clés: optimisation sans dérivées, région de confiance, contraintes de borne, fonctions bruitées Abstract Derivative-free optimization (DFO) has enjoyed renewed interest over the past years, mostly motivated by the ever growing need to solve optimization problems defined by functions whose values are computed by simulation (e.g. engineering design, medical image restoration or groundwater supply). In the last few years, a number of derivative-free optimization methods have been developed and especially model-based trust-region methods have been shown to perform well. In this thesis, we present a new interpolation-based trust-region algorithm which shows to be efficient and globally convergent (in the sense that its convergence is guaranteed to a stationary point from arbitrary starting points). The new algorithm relies on the technique of self-correcting geometry proposed by Scheinberg and Toint [128] in 2009. In their theory, they advanced the understanding of the role of geometry in model-based DFO methods, in our work, we improve the efficiency of their method while maintaining its good theoretical convergence properties. We further examine the influence of different types of interpolation models on the performance of the new algorithm. Furthermore, we extended this method to handle bound constraints by applying an active- set strategy. Considering an active-set method in bound-constrained model-based optimization creates the opportunity of saving a substantial amount of function evaluations. It allows to maintain smaller interpolation sets while proceeding optimization in lower dimensional sub- spaces. The resulting algorithm is shown to be numerically highly competitive. We present results on a test set of smooth problems from the CUTEr collection and compare to well-known state-of-the-art packages from different classes of DFO methods. To report numerical experiments incorporating noise, we create a test set of noisy problems by adding perturbations to the set of smooth problems. The choice of noisy problems was guided by a desire to mimic simulation-based optimization problems. Finally, we will present results on a real-life application of a wing-shape design problem provided by Airbus. Keywords: derivative-free optimization, trust-region method, bound constraints, noisy problems This dissertation would not have been possible without the sup- port of many individuals. First of all, I want to thank my advi- sors, Serge Gratton and Philippe L. Toint, for four years of teach- ing, mentoring, and challenging me. I appreciated very much to work with them. I thank the referees, Luis Nunes Vicente and Jean-Charles Gilbert, and the other members of the jury, Bijan Mohammadi and Matthieu Meaux, for accepting to evaluate my work. The people at CERFACS, and in particular the members of the Algo team, have all been very pleasant to work and learn with, and I thank all of them for a wonderful graduate experience. Es- pecially, I want to mention Selime who is the new sunshine of the Algo team. She has always been very helpful and I wish that one day I can help her as much as she did. I also want to thank the people from the CFD team of CERFACS, in particular Jean-Francois Boussuge and Julien Laurenceau, and people from Airbus France, in particular Pascal Larrieu and Matthieu Meaux for their support and for helpful discussions concerning the use of optimization software in aircraft design. I would like to thank my parents, Marlies and Jörg Tröltzsch, for supporting me during all my projects and for giving me the freedom to find my own way. Moreover, I want to thank my father for proofreading this thesis. Last but not least, I want to thank Christian wholeheartedly for his patience, support and constant encouragement. Contents 1 Introduction 1 2 Model-based derivative-free optimization 5 2.1 Basic material . 6 2.1.1 The trust-region framework . 7 2.1.2 Polynomial interpolation . 8 2.1.3 Lagrange polynomials and Λ-poisedness . 8 2.1.4 Condition number as a measure of well-poisedness . 11 2.2 Polynomial interpolation and regression models . 12 2.2.1 Interpolation sets of variable size . 12 2.2.2 Sub-basis model . 13 2.2.3 Minimum ℓ2-norm model . 14 2.2.4 Minimum Frobenius-norm model . 14 2.2.5 Least-change Frobenius-norm model . 15 2.2.6 Minimum ℓ1-norm model . 16 2.2.7 Least-squares regression model . 16 2.3 A basic interpolation-based trust-region approach . 17 2.4 Self-correcting geometry - a recent approach . 20 2.4.1 The algorithm . 20 2.4.2 Convergence theory . 20 2.5 A modified algorithm with self-correcting geometry . 24 2.5.1 The algorithm . 24 2.5.2 Modifications . 24 2.5.3 Global convergence . 26 2.5.4 Outlook . 30 3 A bound-constrained DFO algorithm 31 3.1 A recursive active-set trust-region algorithm . 32 3.1.1 Outline of the algorithm . 32 3.1.2 Ensuring suitability of a tentative interpolation set . 32 3.1.3 Recursive call in the subspace Sk ....................... 35 3.1.4 Local solver . 37 3.1.5 Defining the next iterate . 37 3.1.6 Avoid re-entering a subspace . 40 3.2 Theoretical issues . 41 3.2.1 Global convergence .
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