Cardinality constrained optimization problems (CCOP) are a new class of optimization problems with many applications. In this thesis, we propose a framework called mathematical programs with disjunctive subspaces constraints (MPDSC), a special case of mathematical programs with disjunctive constraints (MPDC), to investigate CCOP. Our method is different from the relaxed complementarity-type reformulation in the literature. The first contribution of this thesis is that we study various stationarity conditions for MPDSC, and then apply them to CCOP. In particular, we recover disjunctive-type strong (S-) stationarity and Mordukhovich (M-) stationarity for CCOP, and then reveal the relationship between them and those from the relaxed complementarity-type reformulation. The second contribution of this thesis is that we obtain some new results for MPDSC, which do not hold for MPDC in general. We show that many constraint qualifications like the relaxed constant positive linear dependence (RCPLD) coincide with their piecewise versions for MPDSC. Based on such result, we prove that RCPLD implies error bounds for MPDSC. These two results also hold for CCOP. All of these disjunctive-type constraint qualifications for CCOP derived from MPDSC are weaker than those from the relaxed complementarity-type reformulation in some sense. / Graduate

Der Schwerpunkt der vorliegenden Arbeit liegt auf Methoden zur Lösung nichtlinearer Optimierungsprobleme mit der Anforderung, jedes globale Optimum garantiert zu finden und mit einer im Voraus festgesetzten Genauigkeit zu approximieren. Eng verbunden mit dieser deterministischen Optimierung ist die Berechnung von Schranken für den Wertebereich einer Funktion über einem gegebenen Hyperquader. Verschiedene Ansätze, z. B. auf Basis der Intervallarithmetik, werden vorgestellt und analysiert. Im Besonderen werden Methoden zur Schrankengenerierung für multivariate ganz- und gebrochenrationale Polynome mit Hilfe der Darstellung in der Basis der Bernsteinpolynome weiterentwickelt. Weiterhin werden in der Arbeit schrittweise die Bausteine eines deterministischen Optimierungsverfahrens unter Verwendung der berechneten Wertebereichsschranken beschrieben und Besonderheiten für die Optimierung polynomialer Aufgaben näher untersucht. Die Analyse und Bearbeitung einer Aufgabenstellung aus dem Entwicklungsprozess für Fahrzeuggetriebe zeigt, wie die erarbeiteten Ansätze zur Lösung nichtlinearer Optimierungsprobleme die Suche nach energieeffizienten Getrieben mit einer optimalen Struktur unterstützen kann. Kontakt zum Autor: [Nachname] [.] [Vorname] [@] gmx [.] de

We propose a new method to perform functional principal component analysis (FPCA) for discretely observed functional data by solving successive optimization problems. The new framework can be applied to both regularly and irregularly observed data, and to both dense and sparse data. Our method does not require estimates of the individual sample functions or the covariance functions. Hence, it can be used to analyze functional data with multidimensional arguments (e.g. random surfaces). Furthermore, it can be applied to many processes and models with complicated or nonsmooth covariance functions. In our method, smoothness of eigenfunctions is controlled by directly imposing roughness penalties on eigenfunctions, which makes it more efficient and flexible to tune the smoothness. Efficient algorithms for solving the successive optimization problems are proposed. We provide the existence and characterization of the solutions to the successive optimization problems. The consistency of our method is also proved. Through simulations, we demonstrate that our method performs well in the cases with smooth samples curves, with discontinuous sample curves and nonsmooth covariance and with sample functions having two dimensional arguments (random surfaces), repectively. We apply our method to classification problems of retinal pigment epithelial cells in eyes of mice and to longitudinal CD4 counts data. In the second part of this dissertation, we propose a sparse Fisher’s discriminant analysis method with thresholded linear constraints. Various regularized linear discriminant analysis (LDA) methods have been proposed to address the problems of the LDA in high-dimensional settings. Asymptotic optimality has been established for some of these methods when there are only two classes. A difficulty in the asymptotic study for the multiclass classification is that for the two-class classification, the classification boundary is a hyperplane and an explicit formula for the classification error exists, however, in the case of multiclass, the boundary is usually complicated and no explicit formula for the error generally exists. Another difficulty in proving the asymptotic consistency and optimality for sparse Fisher’s discriminant analysis is that the covariance matrix is involved in the constraints of the optimization problems for high order components. It is not easy to estimate a general high-dimensional covariance matrix. Thus, we propose a sparse Fisher’s discriminant analysis method which avoids the estimation of the covariance matrix, provide asymptotic consistency results and the corresponding convergence rates for all components. To prove the asymptotic optimality, we provide an asymptotic upper bound for a general linear classification rule in the case of muticlass which is applied to our method to obtain the asymptotic optimality and the corresponding convergence rate. In the special case of two classes, our method achieves the same as or better convergence rates compared to the existing method. The proposed method is applied to multivariate functional data with wavelet transformations.

Le but de cette thèse est de proposer des méthodes générales afin de contourner l’intractabilité de problèmes d’optimisation multi-objectifs.Dans un premier temps, nous essayons d’apprécier la portée de cette intractabilité en déterminant une borne supérieure, facilement calculable, sur le nombre de points non dominés, connaissant le nombre de valeurs prises par chaque critère.Nous nous attachons ensuite à produire des représentations discrètes et tractables de l’ensemble des points non dominés de toute instance de problèmes d’optimisation multi-objectifs. Ces représentations doivent satisfaire des conditions de couverture, i.e. fournir une bonne approximation, de cardinalité, i.e. ne pas contenir trop de points, et si possible de stabilité, i.e. ne pas contenir de redondances. En s’inspirant de travaux visant à produire des ensembles ε-Pareto de petite taille, nous proposons tout d’abord une extension directe de ces travaux, puis nous axons notre recherche sur des ensembles ε-Pareto satisfaisant une condition supplémentaire de stabilité. Formellement, nous considérons des ensembles ε-Pareto particuliers, appelés (ε, ε′)-noyaux, qui satisfont une propriété de stabilité liée à ε′. Nous établissons des résultats généraux sur les (ε, ε′)-noyaux puis nous proposons des algorithmes polynomiaux qui produisent des (ε, ε′)-noyaux de petite taille pour le cas bi-objectif et nous donnons des résultats négatifs pour plus de deux objectifs. / The goal of this thesis is to propose new general methods to get around the intractability of multi-objective optimization problems.First, we try to give some insight on this intractability by determining an, easily computable, upper bound on the number of nondominated points, knowing the number of values taken on each criterion. Then, we are interested in producingsome discrete and tractable representations of the set of nondominated points for each instance of multi-objective optimization problems. These representations must satisfy some conditions of coverage, i.e. providing a good approximation, cardinality, i.e. it does not contain too many points, and if possible spacing, i.e. it does not include any redundancies. Starting from works aiming to produce ε-Pareto sets of small size, we first propose a direct extension of these works then we focus our research on ε-Pareto sets satisfying an additional condition of stability. Formally, we consider special ε-Pareto sets, called (ε, ε′)-kernels, which satisfy a property of stability related to ε′. We give some general results on (ε, ε′)-kernels and propose some polynomial time algorithms that produce small (ε, ε′)-kernels for the bicriteria case and we give some negative results for the tricriteria case and beyond.

Cette thèse, présente des contributions à la résolution (sur les GPUs) de problèmes d'optimisations réels de grandes tailles. Les problèmes de tournées de véhicules (VRP) et ceux de localisation des hubs (HLP) sont traités. Diverses approches et leur implémentions sur GPU pour résoudre des variantes du VRP sont présentées. Un algorithme génétique (GA) parallèle sur GPU est proposé pour résoudre différentes variantes du HLP. Le GA adapte son codage, sa solution initiale, ses opérateurs génétiques et son implémentation à chacune des variantes traitées. Enfin, nous avons utilisé le GA pour résoudre le HLP avec des incertitudes sur les données.Les tests numériques montrent que les approches proposées exploitent efficacement la puissance de calcul du GPU et ont permis de résoudre de larges instances jusqu'à 6000 nœuds. / This thesis presents contributions to the resolution (on GPUs) of real optimization problems of large sizes. The vehicle routing problems (VRP) and the hub location problems (HLP) are treated. Various approaches implemented on GPU to solve variants of the VRP. A parallel genetic algorithm (GA) on GPU is proposed to solve different variants of the HLP. The proposed GA adapts its encoding, initial solution, genetic operators and its implementation to each of the variants treated. Finally, we used the GA to solve the HLP with uncertainties on the data.The numerical tests show that the proposed approaches effectively exploit the computing power of the GPU and have made it possible to resolve large instances up to 6000 nodes.

im Postscript-Format / PostScript

The presented thesis discusses bilevel programming problems with the focus on solution algorithms. Bilevel programming problem is a hierarchical programming problem, where constraints contain another programming problem. We formulate basic bilevel optimization theory and describe three types of so- lution algorithms for bilevel programming problems: Algorithms based on KKT reformulation where the lower level is replaced by its KKT conditions, algorithms based on optimal value function where the bilevel programming problem is re- duced to a single level problem using the optimal value function of the lower level problem, and algorithms solving linear bilevel programming problems. Using real data for portfolio optimization bilevel programming problems, we compare ability to solve the problems and computing time of some of the pre- sented algorithms. 1

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2016-08-10T13:09:08Z No. of bitstreams: 2 Dissertação - Adriana Batista da Silva - 2016.pdf: 3286914 bytes, checksum: 7be43fd6757af8224a3358e36202f297 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-08-10T13:12:09Z (GMT) No. of bitstreams: 2 Dissertação - Adriana Batista da Silva - 2016.pdf: 3286914 bytes, checksum: 7be43fd6757af8224a3358e36202f297 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-10T13:12:09Z (GMT). No. of bitstreams: 2 Dissertação - Adriana Batista da Silva - 2016.pdf: 3286914 bytes, checksum: 7be43fd6757af8224a3358e36202f297 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-04-29 / Financiadora de Estudos e Projetos- Finep / Linear Programming is an operational search tool applied to problem solving aimed at the optimization of a study system. This study aims to show some applications of linear programming and how it can be used as a motivating teaching skills of Mathematics in High school. We will cover two methods of linear programming problems resolution, Graphics resolution method and the algebraic solution way, which is the Simple Method. We so decided to solve under the assistance of these two methods quoted, optimization problems with two or three variables, since the problems with multiple variables do not match the curriculum of mathematics for high school. We suggest the use of two free applications for mobile phones, to solve optimization problems with several variables, in order to contribute in a di erent approach, seeking to boost in students, interest in the study of mathematics. / A Programação Linear é uma ferramenta da Pesquisa Operacional aplicada à solução de problemas que objetivam a otimização de um sistema de estudo. O presente trabalho busca mostrar algumas aplicações da Programação Linear e como ela pode ser utilizada como elemento motivador no ensino da Matemática no Ensino Médio. Abordaremos dois métodos de resolução de problemas de Programação Linear, o método de Resolução Grá ca e a forma algébrica de solução que é o Método Simplex. Vamos solucionar com auxílio desses dois métodos citados, problemas de otimização com duas e três variáveis, já que problemas com várias variáveis não se adequam a matriz curricular de Matemática para o Ensino Médio. Sugerimos a utilização de dois aplicativos gratuitos para celulares, que solucionam problemas de otimização com várias variáveis como forma de contribuir com uma abordagem diferente, visando despertar no aluno, o interesse pelo estudo da Matemática.

Many real-world decision processes require solving optimization problems which may involve different types of constraints such as inequality and equality constraints. The hurdles in solving these Constrained Optimization Problems (COPs) arise from the challenge of searching a huge variable space in order to locate feasible points with acceptable solution quality. Over the last decades Evolutionary Algorithms (EAs) have brought a tremendous advancement in the area of computer science and optimization with their ability to solve various problems. However, EAs have inherent difficulty in dealing with constraints when solving COPs. This thesis presents a new Agent-based Memetic Algorithm (AMA) for solving COPs, where the agents have the ability to independently select a suitable Life Span Learning Process (LSLP) from a set of LSLPs. Each agent represents a candidate solution of the optimization problem and tries to improve its solution through cooperation with other agents. Evolutionary operators consist of only crossover and one of the self-adaptively selected LSLPs. The performance of the proposed algorithm is tested on benchmark problems, and the experimental results show convincing performance. The quality of individuals in the initial population influences the performance of evolutionary algorithms, especially when the feasible region of the constrained optimization problems is very tiny in comparison to the entire search space. This thesis proposes a method that improves the quality of randomly generated initial solutions by sacrificing very little in diversity of the population. The proposed Search Space Reduction Technique (SSRT) is tested using five different existing EAs, including AMA, by solving a number of state-of-the-art test problems and a real world case problem. The experimental results show SSRT improves the solution quality, and speeds up the performance of the algorithms. The handling of equality constraints has long been a difficult issue for evolutionary optimization methods, although several methods are available in the literature for handling functional constraints. In any optimization problems with equality constraints, to satisfy the condition of feasibility and optimality the solution points must lie on each and every equality constraint. This reduces the size of the feasible space and makes it difficult for EAs to locate feasible and optimal solutions. A new Equality Constraint Handling Technique (ECHT) is presented in this thesis, to enhance the performance of AMA in solving constrained optimization problems with equality constraints. The basic concept is to reach a point on the equality constraint from its current position by the selected individual solution and then explore on the constraint landscape. The technique is used as an agent learning process in AMA. The experimental results confirm the improved performance of the proposed algorithm. This thesis also proposes a Modified Genetic Algorithm (MGA) for solving COPs with equality constraints. After achieving inspiring performance in AMA when dealing with equality constraints, the new technique is used in the design of MGA. The experimental results show that the proposed algorithm overcomes the limitations of GA in solving COPs with equality constraints, and provides good quality solutions.

Submitted by Livia Mello (liviacmello@yahoo.com.br) on 2016-10-11T18:51:01Z No. of bitstreams: 1 DissLSC.pdf: 11398801 bytes, checksum: cc16fd83109503a52839256a9f8c624c (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T12:18:54Z (GMT) No. of bitstreams: 1 DissLSC.pdf: 11398801 bytes, checksum: cc16fd83109503a52839256a9f8c624c (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T12:19:01Z (GMT) No. of bitstreams: 1 DissLSC.pdf: 11398801 bytes, checksum: cc16fd83109503a52839256a9f8c624c (MD5) / Made available in DSpace on 2016-10-21T12:19:09Z (GMT). No. of bitstreams: 1 DissLSC.pdf: 11398801 bytes, checksum: cc16fd83109503a52839256a9f8c624c (MD5) Previous issue date: 2016-09-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / This paper reports the idealization, planning, construction and implementation of activities for the study of quadratic functions and geometric probability for classes of 9th grade of elementary school. It also presents the analysis of the activities undertaken by pupils and conclusions about the proposed objectives and goals achieved. The main objective of the developed activities is to provide students a better learning content covered and issues through a playful approach, interactive and motivating. The specific objectives are to develop the ability to translate a mathematical problem in mathematical language, manipulate algebraic expressions, estimates and comparisons, develop mathematical knowledge as knowing how to express and calculate the area and perimeter of plane figures, calculating probabilities of random events, solve quadratic equations, plotting graphs of quadratic functions and manipulate the software or the GeoGebra application. For this we have created a game of darts adapted and activity sheets containing instructions, questions, tables, graphs, calculation exercises, optimization problems and graphic constructions scripts applied to GeoGebra. The methodology used was the Didactic Engineering. The activities were implemented in two classes of 9th grade of elementary school in two different schools, one class at a school in the municipal school of Bauru, São Paulo, and another class of a school in the state school system the city of Agudos, state of São Paulo. During the application were used 12 50-minute lessons in two classes, with six days of double classes, in which students actively participated in all activities. Our work makes reference to the National Curriculum Parameters (PCN) and other documents governing the teaching of mathematics in public schools in Brazil. We recommend and authorize reproduction of these activities for didactic purposes. / Esta dissertação relata a idealização, o planejamento, a construção e a aplicação de atividades para o estudo de funções quadráticas e probabilidade geométrica para turmas de 9.o ano do ensino fundamental. Também apresenta as análises das atividades realizadas pelos alunos e as conclusões acerca dos objetivos propostos e dos objetivos alcançados. O objetivo principal das atividades elaboradas é proporcionar aos alunos uma melhor aprendizagem dos conteúdos e temas abordados através de uma abordagem lúdica, interativa e motivadora. Os objetivos específicos são desenvolver a capacidade de traduzir um problema matemático na linguagem matemática, manipular expressões algébricas, fazer estimativas e comparações, desenvolver conhecimentos matemáticos como saber expressar e calcular a área e o perímetro de figuras planas, calcular probabilidades de ocorrência de eventos aleatórios, resolver equações quadráticas, traçar gráficos de funções quadráticas e manipular o software ou o aplicativo GeoGebra. Para isto criamos um jogo de dardos adaptado e fichas de atividades contendo instruções, questões, tabelas, gráficos, exercícios de cálculos, problemas de otimização e roteiros de construções gráficas aplicadas ao GeoGebra. A metodologia utilizada neste trabalho foi a Engenharia Didática. As atividades foram aplicadas em duas turmas de 9.o ano do ensino fundamental de duas escolas diferentes, sendo uma turma de uma escola da rede municipal de ensino de Bauru, estado de São Paulo, e outra turma de uma escola da rede estadual de ensino da cidade de Agudos, estado de São Paulo. Durante a aplicação foram utilizadas 12 aulas de 50 minutos nas duas turmas, sendo 6 dias de aulas duplas, nas quais os alunos participaram ativamente de todas as atividades. Nosso trabalho tem como referência os Parâmetros Curriculares Nacionais (PCN) e outros documentos que regem o ensino de matemática nas escolas públicas do Brasil. Recomendamos e autorizamos a reprodução destas atividades para fins didáticos.