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Methods for vector optimization: trust region and proximal on riemannian manifolds and Newton with variable order / Métodos para otimização vetorial: região de confiança e método proximal em variedades riemannianas e método de Newton com ordem variávelPereira, Yuri Rafael Leite 28 August 2017 (has links)
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Previous issue date: 2017-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will analyze three types of method to solve vector optimization problems
in different types of context. First, we will present the trust region method for multiobjective
optimization in the Riemannian context, which retrieves the classical trust region method for
minimizing scalar functions. Under mild assumptions, we will show that each accumulation
point of the generated sequences by the method, if any, is Pareto critical. Next, the proximal
point method for vector optimization and its inexact version will be extended from Euclidean
space to the Riemannian context. Under suitable assumptions on the objective function,
the well-definedness of the methods will be established. Besides, the convergence of any
generated sequence, to a weak efficient point, will be obtained. The last method to be
investigated is the Newton method to solve vector optimization problem with respect to
variable ordering structure. Variable ordering structures are set-valued map with cone values
that to each element associates an ordering. In this analyze we will prove the convergence
of the sequence generated by the algorithm of Newton method and, moreover, we also will
obtain the rate of convergence under variable ordering structures satisfying mild hypothesis. / Neste trabalho, analisaremos três tipos de métodos para resolver problemas de otimização
vetorial em diferentes tipos contextos. Primeiro, apresentaremos o método da Região de
Confiança para resolver problemas multiobjetivo no contexto Riemanniano, o qual recupera o
método da Região de Confiança clássica para minimizar funções escalares. Sob determinadas
suposições, mostraremos que cada ponto de acumulação das sequências geradas pelo método, se houver, é Pareto crítico. Em seguida, o método do ponto proximal para otimização vetorial e sua versão inexata serão estendidos do espaço Euclidiano para o contexto Riemanniano. Sob adequados pressupostos sobre a função objetiva, a boas definições dos métodos serão estabelecidos. Além disso, a convergência de qualquer sequência gerada, para um ponto fracamente eficiente, é obtida. O último método a ser investigado é o método de Newton para resolver o problema de otimização vetorial com respeito a estruturas de ordem variável. Estruturas de ordem variável são aplicações ponto-conjunto cujas imagens são cones que para cada elemento associa uma ordem. Nesta análise, provaremos a convergência da sequência gerada pelo algoritmo do método de Newton e, além disso, também obteremos a taxa de convergência sob estruturas de ordem variável satisfazendo adequadas hipóteses.
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Bayesian Inspired Multi-Fidelity Optimization with Aerodynamic Design ApplicationFischer, Christopher Corey 28 May 2021 (has links)
No description available.
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Revisiting optimization algorithms for maximum likelihood estimationMai, Anh Tien 12 1900 (has links)
Parmi les méthodes d’estimation de paramètres de loi de probabilité en statistique, le
maximum de vraisemblance est une des techniques les plus populaires, comme, sous des conditions l´egères, les estimateurs ainsi produits sont consistants et asymptotiquement efficaces. Les problèmes de maximum de vraisemblance peuvent être traités comme des problèmes de programmation non linéaires, éventuellement non convexe, pour lesquels deux grandes classes de méthodes de résolution sont les techniques de région de confiance et les méthodes de recherche linéaire. En outre, il est possible d’exploiter la structure de ces problèmes pour tenter d’accélerer la convergence de ces méthodes, sous certaines hypothèses. Dans ce travail, nous revisitons certaines approches classiques ou récemment d´eveloppées en optimisation non linéaire, dans le contexte particulier de l’estimation de maximum de vraisemblance. Nous développons également de nouveaux algorithmes pour résoudre ce problème, reconsidérant différentes techniques d’approximation de hessiens, et proposons de nouvelles méthodes de calcul de pas, en particulier dans le cadre des algorithmes de recherche linéaire. Il s’agit notamment d’algorithmes nous permettant de changer d’approximation de hessien et d’adapter la longueur du pas dans une direction de recherche fixée. Finalement, nous évaluons l’efficacité numérique des méthodes proposées dans le cadre de l’estimation de modèles de choix discrets, en
particulier les modèles logit mélangés. / Maximum likelihood is one of the most popular techniques to estimate the parameters
of some given distributions. Under slight conditions, the produced estimators are
consistent and asymptotically efficient. Maximum likelihood problems can be handled
as non-linear programming problems, possibly non convex, that can be solved for instance using line-search methods and trust-region algorithms. Moreover, under some
conditions, it is possible to exploit the structures of such problems in order to speedup
convergence. In this work, we consider various non-linear programming techniques,
either standard or recently developed, within the maximum likelihood estimation perspective. We also propose new algorithms to solve this estimation problem, capitalizing on Hessian approximation techniques and developing new methods to compute steps, in particular in the context of line-search approaches. More specifically, we investigate methods that allow us switching between Hessian approximations and adapting the step length along the search direction. We finally assess the numerical efficiency of the proposed methods for the estimation of discrete choice models, more precisely mixed logit models.
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Revisiting optimization algorithms for maximum likelihood estimationMai, Anh Tien 12 1900 (has links)
Parmi les méthodes d’estimation de paramètres de loi de probabilité en statistique, le
maximum de vraisemblance est une des techniques les plus populaires, comme, sous des conditions l´egères, les estimateurs ainsi produits sont consistants et asymptotiquement efficaces. Les problèmes de maximum de vraisemblance peuvent être traités comme des problèmes de programmation non linéaires, éventuellement non convexe, pour lesquels deux grandes classes de méthodes de résolution sont les techniques de région de confiance et les méthodes de recherche linéaire. En outre, il est possible d’exploiter la structure de ces problèmes pour tenter d’accélerer la convergence de ces méthodes, sous certaines hypothèses. Dans ce travail, nous revisitons certaines approches classiques ou récemment d´eveloppées en optimisation non linéaire, dans le contexte particulier de l’estimation de maximum de vraisemblance. Nous développons également de nouveaux algorithmes pour résoudre ce problème, reconsidérant différentes techniques d’approximation de hessiens, et proposons de nouvelles méthodes de calcul de pas, en particulier dans le cadre des algorithmes de recherche linéaire. Il s’agit notamment d’algorithmes nous permettant de changer d’approximation de hessien et d’adapter la longueur du pas dans une direction de recherche fixée. Finalement, nous évaluons l’efficacité numérique des méthodes proposées dans le cadre de l’estimation de modèles de choix discrets, en
particulier les modèles logit mélangés. / Maximum likelihood is one of the most popular techniques to estimate the parameters
of some given distributions. Under slight conditions, the produced estimators are
consistent and asymptotically efficient. Maximum likelihood problems can be handled
as non-linear programming problems, possibly non convex, that can be solved for instance using line-search methods and trust-region algorithms. Moreover, under some
conditions, it is possible to exploit the structures of such problems in order to speedup
convergence. In this work, we consider various non-linear programming techniques,
either standard or recently developed, within the maximum likelihood estimation perspective. We also propose new algorithms to solve this estimation problem, capitalizing on Hessian approximation techniques and developing new methods to compute steps, in particular in the context of line-search approaches. More specifically, we investigate methods that allow us switching between Hessian approximations and adapting the step length along the search direction. We finally assess the numerical efficiency of the proposed methods for the estimation of discrete choice models, more precisely mixed logit models.
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Μαθηματικές μέθοδοι βελτιστοποίησης προβλημάτων μεγάλης κλίμακας / Mathematical methods of optimization for large scale problemsΑποστολοπούλου, Μαριάννα 21 December 2012 (has links)
Στην παρούσα διατριβή μελετάμε το πρόβλημα της βελτιστοποίησης μη γραμμικών συναρτήσεων πολλών μεταβλητών, όπου η αντικειμενική συνάρτηση είναι συνεχώς διαφορίσιμη σε ένα ανοιχτό υποσύνολο του Rn. Αναπτύσσουμε μαθηματικές μεθόδους βελτιστοποίησης αποσκοπώντας στην επίλυση προβλημάτων μεγάλης κλίμακας, δηλαδή προβλημάτων των οποίων οι μεταβλητές είναι πολλές χιλιάδες, ακόμα και εκατομμύρια. Η βασική ιδέα των μεθόδων που αναπτύσσουμε έγκειται στη θεωρητική μελέτη των χαρακτηριστικών μεγεθών των Quasi-Newton ενημερώσεων ελάχιστης και μικρής μνήμης. Διατυπώνουμε θεωρήματα αναφορικά με το χαρακτηριστικό πολυώνυμο, τον αριθμό των διακριτών ιδιοτιμών και των αντίστοιχων ιδιοδιανυσμάτων. Εξάγουμε κλειστούς τύπους για τον υπολογισμό των ανωτέρω ποσοτήτων, αποφεύγοντας τόσο την αποθήκευση όσο και την παραγοντοποίηση πινάκων. Τα νέα θεωρητικά απoτελέσματα εφαρμόζονται αφενός μεν στην επίλυση μεγάλης κλίμακας υποπροβλημάτων περιοχής εμπιστοσύνης, χρησιμοποιώντας τη μέθοδο της σχεδόν ακριβούς λύσης, αφετέρου δε, στην καμπυλόγραμμη αναζήτηση, η οποία χρησιμοποιεί ένα ζεύγος κατευθύνσεων μείωσης, την Quasi-Newton κατεύθυνση και την κατεύθυνση αρνητικής καμπυλότητας. Η νέα μέθοδος μειώνει δραστικά τη χωρική πολυπλοκότητα των γνωστών αλγορίθμων του μη γραμμικού προγραμματισμού, διατηρώντας παράλληλα τις καλές ιδιότητες σύγκλισής τους. Ως αποτέλεσμα, οι προκύπτοντες νέοι αλγόριθμοι έχουν χωρική πολυπλοκότητα Θ(n). Τα αριθμητικά αποτελέσματα δείχνουν ότι οι νέοι αλγόριθμοι είναι αποδοτικοί, γρήγοροι και πολύ αποτελεσματικοί όταν χρησιμοποιούνται στην επίλυση προβλημάτων με πολλές μεταβλητές. / In this thesis we study the problem of minimizing nonlinear functions of several variables, where the objective function is continuously differentiable on an open subset of Rn. We develop mathematical optimization methods for solving large scale problems, i.e., problems whose variables are many thousands, even millions. The proposed method is based on the theoretical study of the properties of minimal and low memory Quasi-Newton updates. We establish theorems concerning the characteristic polynomial, the number of distinct eigenvalues and corresponding eigenvectors. We derive closed formulas for calculating these quantities, avoiding both the storage and factorization of matrices. The new theoretical results are applied in the large scale trust region subproblem for calculating nearly exact solutions as well as in a curvilinear search that uses a Quasi-Newton and a negative curvature direction. The new method is drastically reducing the spatial complexity of known algorithms of nonlinear programming. As a result, the new algorithms have spatial complexity Θ(n), while they are maintaining good convergence properties. The numerical results show that the proposed algorithms are efficient, fast and very effective when used in solving large scale problems.
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