No description available.

La construction de modèles qui protègent contre les incertitudes dans les données, telles que la variabilité de l'information et l'imprécision est une des principales préoccupations en optimisation sous incertitude. L'incertitude peut affecter différentes domaines, comme le transport, les télécommunications, la finance, etc., ainsi que les différentes parts d'un problème d'optimisation, comme les coefficients de la fonction objectif et /ou les contraintes. De plus, l'ensemble des données incertaines peut être modélisé de différentes façons, comme sous ensembles compactes et convexes de l´espace réel de dimension n, polytopes, produits Cartésiens des intervalles, ellipsoïdes, etc. Une des approches possibles pour résoudre des tels problèmes est de considérer les versions minimax regret, pour lesquelles résoudre un problème sous incertitude revient à trouver une solution qui s'écarte le moins possible de la valeur solution optimale dans tout les cas. Dans le cas des incertitudes définies par intervalles, les versions minimax regret de nombreux problèmes combinatoires polynomiaux sont NP-difficiles, d'ou l'importance d'essayer de réduire l'espace des solutions. Dans ce contexte, savoir quand un élément du problème, représenté par une variable, fait toujours ou jamais partie d'une solution optimal pour toute réalisation des données (variables 1-persistentes et 0-persistentes respectivement), constitue une manière de réduire la taille du problème. Un des principaux objectifs de cette thèse est d'étudier ces questions pour quelques problèmes d'optimisation combinatoire sous incertitude. Nous étudions les versions minimax regret du problème du choix de p éléments parmi m, de l'arbre couvrant minimum et des deux problèmes de plus court chemin. Pour de tels problèmes, dans le cas des incertitudes définis par intervalles, nous étudions le problème de trouver les variables 1- et 0-persistentes. Nous présentons une procédure de pre-traitement du problème, lequel réduit grandement la taille des formulations des versions de minimax regret. Nous nous intéressons aussi à la version minimax regret du problème de programmation linéaire dans le cas où les coefficients de la fonction objectif sont incertains et l'ensemble des données incertaines est polyédral. Dans le cas où l'ensemble des incertitudes est défini par des intervalles, le problème de trouver le regret maximum est NP-difficile. Nous présentons des cas spéciaux ou les problèmes de maximum regret et de minimax regret sont polynomiaux. Dans le cas où l´ensemble des incertitudes est défini par un polytope, nous présentons un algorithme pour trouver une solution exacte au problème de minimax regret et nous discutons les résultats numériques obtenus dans un grand nombre d´instances générées aléatoirement. Nous étudions les relations entre le problème de 1-centre continu et la version minimax regret du problème de programmation linéaire dans le cas où les coefficients de la fonction objectif sont évalués à l´aide des intervalles. En particulier, nous décrivons la géométrie de ce dernier problème, nous généralisons quelques résultats en théorie de localisation et nous donnons des conditions sous lesquelles certaines variables peuvet être éliminées du problème. Finalement, nous testons ces conditions dans un nombre d´instances générées aléatoirement et nous donnons les conclusions.

The diploma thesis deals with ways of teaching combinatorics at a secondary school. Specifically, I analyzed selected mathematics textbooks for secondary schools in terms of introducing concepts and operations of combinatorics and in terms of types of tasks used. I carried out interviews with six secondary school mathematics teachers and observations of their lessons in order to describe their method of teaching combinatorics. Using results of tests written by these teachers' pupils, I examined whether and how their solving strategies and errors might be influenced by their teachers' approach to teaching combinatorics. Finally, I compared my results with the existing results of mathematics education research on pupils' combinatorial reasoning. The work is divided into four chapters; the first three are theoretical (curricular documents for selected schools, analysis of textbooks on combinatorics in terms of the implementation of combinatorial concepts and operations, selected research about pupils' solving strategies and errors for combinatorial problems, methods of checking the correctness of their solutions. and the impact of ways of teaching combinatorics on pupils' performance). Chapter 4 focuses on my own research which consists of interviews with teachers, observations of lessons on combinatorics, the...

La construction de modèles qui protègent contre les incertitudes dans les données, telles que la variabilité de l'information et l'imprécision est une des principales préoccupations en optimisation sous incertitude. L'incertitude peut affecter différentes domaines, comme le transport, les télécommunications, la finance, etc. ainsi que les différentes parts d'un problème d'optimisation, comme les coefficients de la fonction objectif et /ou les contraintes. De plus, l'ensemble des données incertaines peut être modélisé de différentes façons, comme sous ensembles compactes et convexes de l´espace réel de dimension n, polytopes, produits Cartésiens des intervalles, ellipsoïdes, etc.<p><p>Une des approches possibles pour résoudre des tels problèmes est de considérer les versions minimax regret, pour lesquelles résoudre un problème sous incertitude revient à trouver une solution qui s'écarte le moins possible de la valeur solution optimale dans tout les cas. <p><p>Dans le cas des incertitudes définies par intervalles, les versions minimax regret de nombreux problèmes combinatoires polynomiaux sont NP-difficiles, d'ou l'importance d'essayer de réduire l'espace des solutions. Dans ce contexte, savoir quand un élément du problème, représenté par une variable, fait toujours ou jamais partie d'une solution optimal pour toute réalisation des données (variables 1-persistentes et 0-persistentes respectivement), constitue une manière de réduire la taille du problème. Un des principaux objectifs de cette thèse est d'étudier ces questions pour quelques problèmes d'optimisation combinatoire sous incertitude.<p><p>Nous étudions les versions minimax regret du problème du choix de p éléments parmi m, de l'arbre couvrant minimum et des deux problèmes de plus court chemin. Pour de tels problèmes, dans le cas des incertitudes définis par intervalles, nous étudions le problème de trouver les variables 1- et 0-persistentes. Nous présentons une procédure de pre-traitement du problème, lequel réduit grandement la taille des formulations des versions de minimax regret.<p><p>Nous nous intéressons aussi à la version minimax regret du problème de programmation linéaire dans le cas où les coefficients de la fonction objectif sont incertains et l'ensemble des données incertaines est polyédral. Dans le cas où l'ensemble des incertitudes est défini par des intervalles, le problème de trouver le regret maximum est NP-difficile. Nous présentons des cas spéciaux ou les problèmes de maximum regret et de minimax regret sont polynomiaux. Dans le cas où l´ensemble des incertitudes est défini par un polytope, nous présentons un algorithme pour trouver une solution exacte au problème de minimax regret et nous discutons les résultats numériques obtenus dans un grand nombre d´instances générées aléatoirement.<p><p>Nous étudions les relations entre le problème de 1-centre continu et la version minimax regret du problème de programmation linéaire dans le cas où les coefficients de la fonction objectif sont évalués à l´aide des intervalles. En particulier, nous décrivons la géométrie de ce dernier problème, nous généralisons quelques résultats en théorie de localisation et nous donnons des conditions sous lesquelles certaines variables peuvet être éliminées du problème. Finalement, nous testons ces conditions dans un nombre d´instances générées aléatoirement et nous donnons les conclusions. / Doctorat en sciences, Orientation recherche opérationnelle / info:eu-repo/semantics/nonPublished

L’objectif de ce projet de trois ans est de proposer des avancées conceptuelles et technologiques dans la résolution de problèmes d’ordonnancement du personnel. L’atteinte de cet objectif passe par la proposition de nouveaux algorithmes basés sur les métaheuristiques et leur implémentation sur les architectures de calcul haute performance. Ce projet s’inscrit en complémentarité du projet HORUS qui bénéficie d’une subvention ANR et qui réunit les expertises scientifiques de deux laboratoires universitaires spécialisés en optimisation et en calcul parallèle : l’équipe SysCom du laboratoire CReSTIC de l’URCA et l’équipe CaRO du laboratoire PRiSM de l’UVSQ. Les avancées technologiques proposées s’appuient également sur les moyens de calcul haute performance offerts par le Centre de Calcul Régional Champagne-Ardenne. / .Many academic and industrial optimization problems are multi-objective and have been of particular interest to researchers in recent years. These problems usually do not have a single optimal solution but a set of best trade-off solutions which form the so-called Pareto front in the objective space. In order to approximate the Pareto front, multi-objective evolutionary algorithms (MOEAs) have been largely investigated in the fields of continuous and combinatorial optimization. Contrary to some classical algorithms, MOEAs have the ability to provide a number of solutions in one single run and are less sensitive to the shape of the Pareto front.As they often require a high amount of computing resources to explore large portions of the search space and handle complex real-life constraints, MOEAs could greatly benefit from today's high-performance computing architectures. Although significant progress has been made in recent years in the design and improvement of parallel models for evolutionary algorithms, most of these models have limited scalability and ability to solve various problems. In fact, solving multi-objective combinatorial optimization problems efficiently on a large number of processors remains a challenge today.This thesis aims to propose an island model which is based on objective space division. The main features of the proposed model are the following (i) An organizer has a global view of the current search via a global archive (ii) Asynchronous cooperation between islands, especially for the exchange of local archives with the organizer to limit model overheads (iii)Control islands to guide the exploration of the search space and improve diversity (iv) A periodic use of a specific local search procedure to improve convergence. Extensive experiments have been conducted to evaluate the performance of the approach and more particularly of each component in the resolution of two classical combinatorial problems, the travelling salesman problem and quadratic assignment problem. Extensibility and quality of the solutions are analyzed compared to state-of-the-art parallel models.

O trabalho proposto enfoca a obtenção de planos de restabelecimento de energia de forma automática para a reenergização de redes de distribuição de energia elétrica. Assim sendo, consideram-se situações que deixam regiões do sistema sem energia. A interrupção do serviço pode ser causada por falhas no circuito de distribuição ou pela necessidade de isolar zonas do mesmo para serviços de manutenção. O restabelecimento do fornecimento da energia aos consumidores é um problema com múltiplos objetivos, alguns deles conflitantes. O problema de restabelecimento envolve funções cujas características, em geral, dificultam o uso das técnicas de programação matemática para obter planos de restabelecimento. Além disso, as propostas usando tais metodologias de programação são afetadas intensamente pelo problema de explosão combinatória. Os Algoritmos Evolucionários (AEs) têm apresentado resultados animadores para esse problema. Contudo, tais algoritmos ainda apresentam dificuldades para a rápida obtenção de planos de restabelecimento para redes de tamanho real (grande porte) de maneira a tornar possível sua aplicação em tempo real. Este trabalho propõe uma nova técnica baseada em AEs para o problema de restabelecimento. Essa proposta envolve também um nova forma de representar computacionalmente as redes de distribuição e de modificar a configuração das mesmas. Também é proposto um fluxo de carga específico para ser utilizado com a nova representação. Diversos testes são apresentados utilizando redes com diferentes tamanhos, no intuito de se avaliar a potencialidade da técnica proposta. / The proposed research focus on the automatic elaboration of plans for service restoration in electrical distribution systems. So this research considers situations that leave network regions out-of-service. The service interruption may be caused by faults in the distribution circuit or by isolation of circuit zones for maintenance task. The restoration of the energy supply to the consumers is a multiobjective problem, with a certain degree of conflict. The restoration problem considers functions whose characteristics, in general, difficult the use of mathematical programming techniques to obtain restoration plans. Moreover, the approaches using such functions are intensively affected by the combinatorial explosion problem. The Evolutionary Algorithms have shown relevant results for this problem. Nevertheless, these techniques still have difficulties to obtain restoration plans, in a fast manner, for real size networks (i.e., large size). This work proposes a new methodology based on Evolutionary Algorithms. This approach involves also a new way to computationally represent distribution networks and to modify the network configurations. A specific load flow to be used with the new representation is also proposed. Severals tests are shown, using networks with different sizes, to evaluate the potential of the proposed technique.

Made available in DSpace on 2015-03-03T15:36:09Z (GMT). No. of bitstreams: 1 JoseRAB_DISSERT.pdf: 776491 bytes, checksum: bef691e2a550b6345b490b668bd8cb38 (MD5) Previous issue date: 2013-02-22 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / Humans, as well as some animals are born gifted with the ability to perceive quantities. The needs that came from the evolution of societies and technological resources make the the optimization of such counting methods necessary. Although necessary and useful, there are a lot of diculties in the teaching of such methods.In order to broaden the range of available tools to teach Combinatorial Analysis, a owchart is presented in this work with the goal of helping the students to x the initial concepts of such subject via pratical exercises / Os seres humanos, assim como alguns animais, nascem dotados da capacidade de perceber quantidades. Portanto t?cnicas para contar quantidades foi um passo natural no desenvolvimento do homem. As necessidades provindas da evolu??o das sociedades e recursos tecnol?gicos tornam necess?rio a otimiza??o de tais m?todos de contagem. Apesar de necess?rio e ?til, o estudo desses m?todos no Ensino M?dio esbarram em dificuldades did?ticas. Com o objetivo de ampliar o leque de ferramentas dispon?veis aos professores para o ensino de An?lise Combinat?ria apresentamos neste trabalho um fluxograma que pretende dinamizar o processo de fixa??o dos conceito via resolu??o de exerc?cios

O trabalho proposto enfoca a obtenção de planos de restabelecimento de energia de forma automática para a reenergização de redes de distribuição de energia elétrica. Assim sendo, consideram-se situações que deixam regiões do sistema sem energia. A interrupção do serviço pode ser causada por falhas no circuito de distribuição ou pela necessidade de isolar zonas do mesmo para serviços de manutenção. O restabelecimento do fornecimento da energia aos consumidores é um problema com múltiplos objetivos, alguns deles conflitantes. O problema de restabelecimento envolve funções cujas características, em geral, dificultam o uso das técnicas de programação matemática para obter planos de restabelecimento. Além disso, as propostas usando tais metodologias de programação são afetadas intensamente pelo problema de explosão combinatória. Os Algoritmos Evolucionários (AEs) têm apresentado resultados animadores para esse problema. Contudo, tais algoritmos ainda apresentam dificuldades para a rápida obtenção de planos de restabelecimento para redes de tamanho real (grande porte) de maneira a tornar possível sua aplicação em tempo real. Este trabalho propõe uma nova técnica baseada em AEs para o problema de restabelecimento. Essa proposta envolve também um nova forma de representar computacionalmente as redes de distribuição e de modificar a configuração das mesmas. Também é proposto um fluxo de carga específico para ser utilizado com a nova representação. Diversos testes são apresentados utilizando redes com diferentes tamanhos, no intuito de se avaliar a potencialidade da técnica proposta. / The proposed research focus on the automatic elaboration of plans for service restoration in electrical distribution systems. So this research considers situations that leave network regions out-of-service. The service interruption may be caused by faults in the distribution circuit or by isolation of circuit zones for maintenance task. The restoration of the energy supply to the consumers is a multiobjective problem, with a certain degree of conflict. The restoration problem considers functions whose characteristics, in general, difficult the use of mathematical programming techniques to obtain restoration plans. Moreover, the approaches using such functions are intensively affected by the combinatorial explosion problem. The Evolutionary Algorithms have shown relevant results for this problem. Nevertheless, these techniques still have difficulties to obtain restoration plans, in a fast manner, for real size networks (i.e., large size). This work proposes a new methodology based on Evolutionary Algorithms. This approach involves also a new way to computationally represent distribution networks and to modify the network configurations. A specific load flow to be used with the new representation is also proposed. Severals tests are shown, using networks with different sizes, to evaluate the potential of the proposed technique.

<p>In this thesis, we study optimization problems which have a combinatorial aspect to them. Search space for such problems quickly grows large - exponentially - with respect to the problem dimension. Thus, exhaustive search becomes intractable and we need good relaxations to solve combinatorial problems efficiently. Another challenge arises due to the high dimensionality of such problems and lack of large number of samples. Our aim is to come up with innovative approaches that solve the problem in polynomial time and sample complexity. We discuss three combinatorial optimization problems and provide continuous relaxations for them. Our continuous relaxations involve both convex and nonconvex (invex) relaxations. Furthermore, we provide efficient first order algorithms to solve a general class of invex problems with provable convergence rate guarantees. The three combinatorial problems we study in this work are – learning the directed structure of a Bayesian network using blackbox data, fair sparse regression on a biased dataset where bias depends upon a hidden binary attribute and mixed linear regression. We propose convex relaxation for the first problem, while the other two are solved using invex relaxation. On the first problem, we come up with a novel notion of low rank representation of conditional probability tables for a Bayesian network and connect it to Fourier transformation of real valued set functions to recover the exact structure of the Bayesian networks. For the second problem, we propose a novel invex relaxation for the combinatorial version of sparse linear regression with fairness. For the final problem, we again use invex relaxation to learn a mixture of sparse linear regression models. We formally show correctness of our proposed methods and provide provable theoretical guarantees for efficient computational and sample complexity. We also develop efficient first order algorithms to solve invex problems. We provide convergence rate analysis for our proposed methods. Furthermore, we also discuss possible future research directions and the problems we want to tackle in future.</p>

Combinatorial problems are ubiquitous in our society and solving such problems efficiently is often crucial. One technique for solving combinatorial problems is constraint-based local search. Its compositional nature together with its efficiency on large problem instances have made this technique particularly attractive. In this thesis we contribute to simplifying the solving of combinatorial problems using constraint-based local search. To provide higher-level modelling options, we introduce set variables and set constraints in local search by extending relevant local search concepts. We also propose a general scheme to follow in order to define what we call natural and balanced constraint measures, and accordingly define such measures for over a dozen set constraints. However, defining such measures for a new constraint is time-consuming and error-prone. To relieve the user from this, we provide generic measures for any set constraint modelled in monadic existential second-order logic. We also theoretically relate these measures to our proposed general scheme, and discuss implementation issues such as incremental algorithms and their worst-case complexities. To enable higher-level search algorithms, we introduce constraint-directed neighbourhoods in local search by proposing new constraint primitives for representing such neighbourhoods. Based on a constraint, possibly modelled in monadic existential second-order logic, these primitives return neighbourhoods with moves that are known in advance to achieve a decrease (or preservation, or increase) of the constraint measures, without the need to iterate over any other moves. We also present a framework for constraint-based local search where one can model and solve combinatorial problems with set variables and set constraints, use any set constraint modelled in monadic existential second-order logic, as well as use constraint-directed neighbourhoods. Experimental results on three real-life problems show the usefulness in practice of our theoretical results: our running times are comparable to the current state-of-the-art approaches to solving the considered problems.