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Salazar-Neumann, Martha. "Advances in robust combinatorial optimization and linear programming". Doctoral thesis, Universite Libre de Bruxelles, 2010. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210192.
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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.
Doctorat en sciences, Orientation recherche opérationnelle
info:eu-repo/semantics/nonPublished
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Iemets, O. O. y T. M. Barbolina. "Linear-fractional combinatorial optimization problems: model and solving". Thesis, Sumy State University, 2016. http://essuir.sumdu.edu.ua/handle/123456789/46962.
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Euclidean problems of linear-fractional combinatorial optimization on arrangements are discussed. Authors propose the model of practical problem. Also the polynomial algorithm for solving linear-fractional combinatorial optimization problems on arrangements is discussed.
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Cheng, Jianqiang. "Stochastic Combinatorial Optimization". Thesis, Paris 11, 2013. http://www.theses.fr/2013PA112261.
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Dans cette thèse, nous étudions trois types de problèmes stochastiques : les problèmes avec contraintes probabilistes, les problèmes distributionnellement robustes et les problèmes avec recours. Les difficultés des problèmes stochastiques sont essentiellement liées aux problèmes de convexité du domaine des solutions, et du calcul de l’espérance mathématique ou des probabilités qui nécessitent le calcul complexe d’intégrales multiples. A cause de ces difficultés majeures, nous avons résolu les problèmes étudiées à l’aide d’approximations efficaces.Nous avons étudié deux types de problèmes stochastiques avec des contraintes en probabilités, i.e., les problèmes linéaires avec contraintes en probabilité jointes (LLPC) et les problèmes de maximisation de probabilités (MPP). Dans les deux cas, nous avons supposé que les variables aléatoires sont normalement distribués et les vecteurs lignes des matrices aléatoires sont indépendants. Nous avons résolu LLPC, qui est un problème généralement non convexe, à l’aide de deux approximations basée sur les problèmes coniques de second ordre (SOCP). Sous certaines hypothèses faibles, les solutions optimales des deux SOCP sont respectivement les bornes inférieures et supérieures du problème du départ. En ce qui concerne MPP, nous avons étudié une variante du problème du plus court chemin stochastique contraint (SRCSP) qui consiste à maximiser la probabilité de la contrainte de ressources. Pour résoudre ce problème, nous avons proposé un algorithme de Branch and Bound pour calculer la solution optimale. Comme la relaxation linéaire n’est pas convexe, nous avons proposé une approximation convexe efficace. Nous avons par la suite testé nos algorithmes pour tous les problèmes étudiés sur des instances aléatoires. Pour LLPC, notre approche est plus performante que celles de Bonferroni et de Jaganathan. Pour MPP, nos résultats numériques montrent que notre approche est là encore plus performante que l’approximation des contraintes probabilistes individuellement.La deuxième famille de problèmes étudiés est celle relative aux problèmes distributionnellement robustes où une partie seulement de l’information sur les variables aléatoires est connue à savoir les deux premiers moments. Nous avons montré que le problème de sac à dos stochastique (SKP) est un problème semi-défini positif (SDP) après relaxation SDP des contraintes binaires. Bien que ce résultat ne puisse être étendu au cas du problème multi-sac-à-dos (MKP), nous avons proposé deux approximations qui permettent d’obtenir des bornes de bonne qualité pour la plupart des instances testées. Nos résultats numériques montrent que nos approximations sont là encore plus performantes que celles basées sur les inégalités de Bonferroni et celles plus récentes de Zymler. Ces résultats ont aussi montré la robustesse des solutions obtenues face aux fluctuations des distributions de probabilités. Nous avons aussi étudié une variante du problème du plus court chemin stochastique. Nous avons prouvé que ce problème peut se ramener au problème de plus court chemin déterministe sous certaine hypothèses. Pour résoudre ce problème, nous avons proposé une méthode de B&B où les bornes inférieures sont calculées à l’aide de la méthode du gradient projeté stochastique. Des résultats numériques ont montré l’efficacité de notre approche. Enfin, l’ensemble des méthodes que nous avons proposées dans cette thèse peuvent s’appliquer à une large famille de problèmes d’optimisation stochastique avec variables entièresIn this thesis, we studied three types of stochastic problems: chance constrained problems, distributionally robust problems as well as the simple recourse problems. For the stochastic programming problems, there are two main difficulties. One is that feasible sets of stochastic problems is not convex in general. The other main challenge arises from the need to calculate conditional expectation or probability both of which are involving multi-dimensional integrations. Due to the two major difficulties, for all three studied problems, we solved them with approximation approaches.We first study two types of chance constrained problems: linear program with joint chance constraints problem (LPPC) as well as maximum probability problem (MPP). For both problems, we assume that the random matrix is normally distributed and its vector rows are independent. We first dealt with LPPC which is generally not convex. We approximate it with two second-order cone programming (SOCP) problems. Furthermore under mild conditions, the optimal values of the two SOCP problems are a lower and upper bounds of the original problem respectively. For the second problem, we studied a variant of stochastic resource constrained shortest path problem (called SRCSP for short), which is to maximize probability of resource constraints. To solve the problem, we proposed to use a branch-and-bound framework to come up with the optimal solution. As its corresponding linear relaxation is generally not convex, we give a convex approximation. Finally, numerical tests on the random instances were conducted for both problems. With respect to LPPC, the numerical results showed that the approach we proposed outperforms Bonferroni and Jagannathan approximations. While for the MPP, the numerical results on generated instances substantiated that the convex approximation outperforms the individual approximation method.Then we study a distributionally robust stochastic quadratic knapsack problems, where we only know part of information about the random variables, such as its first and second moments. We proved that the single knapsack problem (SKP) is a semedefinite problem (SDP) after applying the SDP relaxation scheme to the binary constraints. Despite the fact that it is not the case for the multidimensional knapsack problem (MKP), two good approximations of the relaxed version of the problem are provided which obtain upper and lower bounds that appear numerically close to each other for a range of problem instances. Our numerical experiments also indicated that our proposed lower bounding approximation outperforms the approximations that are based on Bonferroni's inequality and the work by Zymler et al.. Besides, an extensive set of experiments were conducted to illustrate how the conservativeness of the robust solutions does pay off in terms of ensuring the chance constraint is satisfied (or nearly satisfied) under a wide range of distribution fluctuations. Moreover, our approach can be applied to a large number of stochastic optimization problems with binary variables.Finally, a stochastic version of the shortest path problem is studied. We proved that in some cases the stochastic shortest path problem can be greatly simplified by reformulating it as the classic shortest path problem, which can be solved in polynomial time. To solve the general problem, we proposed to use a branch-and-bound framework to search the set of feasible paths. Lower bounds are obtained by solving the corresponding linear relaxation which in turn is done using a Stochastic Projected Gradient algorithm involving an active set method. Meanwhile, numerical examples were conducted to illustrate the effectiveness of the obtained algorithm. Concerning the resolution of the continuous relaxation, our Stochastic Projected Gradient algorithm clearly outperforms Matlab optimization toolbox on large graphs
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Burer, Samuel A. "New algorithmic approaches for semidefinite programming with applications to combinatorial optimization". Diss., Georgia Institute of Technology, 2001. http://hdl.handle.net/1853/30268.
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Sidford, Aaron Daniel. "Iterative methods, combinatorial optimization, and linear programming beyond the universal barrier". Thesis, Massachusetts Institute of Technology, 2015. http://hdl.handle.net/1721.1/99848.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 256-266).
In this thesis we consider fundamental problems in continuous and combinatorial optimization that occur pervasively in practice and show how to improve upon the best known theoretical running times for solving these problems across a broad range of parameters. Using and improving techniques from diverse disciplines including spectral graph theory, numerical analysis, data structures, and convex optimization we provide the first theoretical improvements in decades for multiple classic problems ranging from linear programming to linear system solving to maximum flow. Key results in this thesis include the following: -- Linear Programming: We provide the first general improvement to both the running time and convergence rate of polynomial time algorithms for solving linear programs in over 15 years. For a linear program with constraint matrix A, with z nonzero entries, and bit complexity L our algorithm runs in time [mathematical formula] -- Directed Maximum Flow: We provide an [mathematical formula] time algorithm for solving the-maximum flow problem on directed graphs with m edges, n vertices, and capacity ratio U improving upon the running time of [mathematical formula] achieved over 15 years ago by Goldberg and Rao. -- Undirected Approximate Flow: We provide one of the first almost linear time algorithms for approximately solving undirected maximum flow improving upon the previous fastest running time by a factor of [mathematical formula] for graphs with n vertices. -- Laplacian System Solvers: We improve upon the previous best known algorithms for solving Laplacian systems in standard unit cost RAM model, achieving a running time of [mathematical formula] for solving a Laplacian system of equations. -- Linear System Solvers: We obtain a faster asymptotic running time than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. * More: We improve the running time for multiple problems including regression, generalized lossy flow, multicommodity flow, and more.
by Aaron Sidford.
Ph. D.
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Björklund, Henrik. "Combinatorial Optimization for Infinite Games on Graphs". Doctoral thesis, Uppsala University, Department of Information Technology, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4751.
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Games on graphs have become an indispensable tool in modern computer science. They provide powerful and expressive models for numerous phenomena and are extensively used in computer- aided verification, automata theory, logic, complexity theory, computational biology, etc.
The infinite games on finite graphs we study in this thesis have their primary applications in verification, but are also of fundamental importance from the complexity-theoretic point of view. They include parity, mean payoff, and simple stochastic games.
We focus on solving graph games by using iterative strategy improvement and methods from linear programming and combinatorial optimization. To this end we consider old strategy evaluation functions, construct new ones, and show how all of them, due to their structural similarities, fit into a unifying combinatorial framework. This allows us to employ randomized optimization methods from combinatorial linear programming to solve the games in expected subexponential time.
We introduce and study the concept of a controlled optimization problem, capturing the essential features of many graph games, and provide sufficent conditions for solvability of such problems in expected subexponential time.
The discrete strategy evaluation function for mean payoff games we derive from the new controlled longest-shortest path problem, leads to improvement algorithms that are considerably more efficient than the previously known ones, and also improves the efficiency of algorithms for parity games.
We also define the controlled linear programming problem, and show how the games are translated into this setting. Subclasses of the problem, more general than the games considered, are shown to belong to NP intersection coNP, or even to be solvable by subexponential algorithms.
Finally, we take the first steps in investigating the fixed-parameter complexity of parity, Rabin, Streett, and Muller games.
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Ferroni, Nicola. "Exact Combinatorial Optimization with Graph Convolutional Neural Networks". Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/17502/.
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Combinatorial optimization problems are typically tackled by the branch-and-bound paradigm. We propose to learn a variable selection policy for branch-and-bound in mixed-integer linear programming, by imitation learning on a diversified variant of the strong branching expert rule. We encode states as bipartite graphs and parameterize the policy as a graph convolutional neural network. Experiments on a series of synthetic problems demonstrate that our approach produces policies that can improve upon expert-designed branching rules on large problems, and generalize to instances significantly larger than seen during training.
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Weltge, Stefan [Verfasser] y Volker [Akademischer Betreuer] Kaibel. "Sizes of linear descriptions in combinatorial optimization / Stefan Weltge. Betreuer: Volker Kaibel". Magdeburg : Universitätsbibliothek, 2015. http://d-nb.info/1082625868/34.
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Wang, Xia. "Applications of genetic algorithms, dynamic programming, and linear programming to combinatorial optimization problems". College Park, Md.: University of Maryland, 2008. http://hdl.handle.net/1903/8778.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2008.Thesis research directed by: Applied Mathematics & Statistics, and Scientific Computation Program. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Chakrabarty, Deeparnab. "Algorithmic aspects of connectivity, allocation and design problems". Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24659.
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Thesis (Ph.D.)--Computing, Georgia Institute of Technology, 2008.Committee Chair: Vazirani, Vijay; Committee Member: Cook, William; Committee Member: Kalai, Adam; Committee Member: Tetali, Prasad; Committee Member: Thomas, Robin
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Mazzieri, Diego. "Machine Learning for combinatorial optimization: the case of Vehicle Routing". Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/24688/.
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The Vehicle Routing Problem (VRP) is one of the most intensively studied combinatorial optimization problems in the Operations Research (OR) community. Its relevance is not only related to the various real-world applications it deals with, but to its inherent complexity being an NP-hard problem.
From its original formulation more than 60 years ago, numerous mathematical models and algorithms have been proposed to solve VRP. The most recent trend is to leverage Machine Learning (ML) in conjunction with these traditional approaches to enhance their performance.
In particular, this work investigates the use of ML-driven components as destroy or repair methods inside the Large Neighborhood Search (LNS) metaheuristic, trying to understand if, where, and when it is effective to apply them in the context of VRP.
For these purposes, we propose NeuRouting, an open-source hybridization framework aimed at facilitating the integration between ML and LNS.
Regarding the destroy phase, we adopt a Graph Neural Network (GNN) assisted heuristic, which we hybridize with a neural repair methodology taken from the literature. We investigate this integration both on its own and as part of an Adaptive Large Neighborhood Search (ALNS), performing an empirical study on instances of various sizes and against some traditional solvers.
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Brandt, Aléx Fernando 1990. "Algoritmos exatos para problemas de dilatação mínima em grafos geométricos". [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/275536.
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Orientadores: Cid Carvalho de Souza, Pedro Jussieu de RezendeDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação
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Resumo: Seja P um conjunto de pontos no plano. O grafo geométrico de P, G(P) = (P, E), é o grafo ponderado completo cujos vértices correspondem aos pontos de P e no qual o custo de uma aresta {i, j} é dado pela distância Euclidiana entre os pontos i e j. Inicialmente, considere um problema genérico em que se quer construir uma rede com boa qualidade de conexão ligando um conjunto de locais representados por pontos no plano. Em muitas aplicações deste tipo, o problema pode ser modelado com o auxílio de um grafo geométrico. Isso ocorre quando, por exemplo, para um par de pontos, a medida de qualidade é definida como a razão entre o comprimento do menor caminho que os conecta na rede projetada e a respectiva distância Euclidiana. Esta razão determina a dilatação daquele par de pontos na rede. Já a dilatação da rede construída em si é dada pela dilatação máxima sobre todos os pares de pontos. Nesta dissertação apresentamos métodos exatos para resolução dos problemas dedicados a encontrar uma árvore geradora ou uma triangulação planar de dilatação mínima, denominados, respectivamente, Problema da Árvore Geradora de Dilatação Mínima (MDSTP) e Problema da Triangulação de Dilatação Mínima (MDTP). Os métodos descritos são compostos principalmente pela formulação, redução e resolução de programas lineares inteiros mistos. A redução do tamanho destes modelos matemáticos é feita explorando-se a geometria dos problemas por meio de rotinas que determinam a presença ou da ausência de certos elementos da formulação em soluções com dilatação menor ou igual ao limitante primal fornecido por uma heurística. A aplicação destas rotinas em uma fase de pré-processamento permite uma redução significativa do tamanho do modelo levando à aceleração do seu tempo de resolução. Com a combinação destas técnicas obteve-se, pela primeira vez, soluções comprovadamente ótimas de instâncias até 20 pontos para o MDSTP e até 70 pontos para o MDTP. Os problemas e suas formulações, bem como suas formas de redução e de resolução, são apresentados em detalhes. Além disso, são feitas análises de desempenho computacional não só dos métodos exatos, mas também de algoritmos propostas por outros autores
Abstract: Let P be a set of points in the plane. The geometric graph of P, G(P) =(P, E), is the complete weighted graph whose vertices correspond to the points of P and in which the cost of an edge {i, j} is given by the Euclidean distance between the points i and j. Initially, consider a general problem where one wants to build a network with good connection quality joining a set of sites represented by points in the plane. In many applications of this kind, the problem can be modeled with the aid of a geometric graph. This occurs when, for example, for a pair of points, the quality measure is defined as the ratio of the length of the shortest path that connects them in the designed network and their Euclidean distance. This ratio determines the dilation of that pair of points in the network. On the other hand, the dilation of the built network itself is given by the maximum dilation over all pair of points. In this dissertation we present exact methods for solving problems dedicated to finding a spanning tree or a planar triangulation of minimum dilation, named, respectively, the Minimum Dilation Spanning Tree Problem (MDSTP) and Minimum Dilation Triangulation Problem (MDTP). The described methods are composed primarily by the formulation, downsizing and resolution of mixed integer linear programs. The downsizing of these mathematical models is done by exploiting the geometry of the problems by means of routines that determine the need of the presence or the absence of certain elements of the formulation in solutions with dilation less than or equal to the primal bound supplied by a heuristic. Applying these routines in a pre-processing phase allows a significant reduction of the model size leading to the acceleration of its resolution time. With the combination of these techniques, for the first time, proven optimal solutions of instances with up to 20 points for the MDSTP and up to 70 points to the MDTP were obtained. The problems and their formulations, as well as ways of reducing and solving them, are presented in detail. Furthermore, analyzes of computational performance not only of the exact methods, but also of algorithms proposed by other authors are made
Mestrado
Ciência da Computação
Mestre em Ciência da Computação
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Klein, Kim-Manuel [Verfasser]. "About the Structure and Sensitivity of Integer Linear Programs and their Application in Combinatorial Optimization / Kim-Manuel Klein". Kiel : Universitätsbibliothek Kiel, 2017. http://d-nb.info/1137867418/34.
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Bogue, Eduardo Theodoro 1990. "O problema da máxima interseção de k-subconjuntos". [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/275507.
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Orientadores: Cid Carvalho de Souza, Eduardo Candido XavierDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação
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Resumo: Neste projeto, nós estudamos o Problema da Máxima Interseção de k-Subconjuntos (kMIS). Dado um inteiro k, um conjunto base U e uma coleção S de subconjuntos de U, o problema kMIS consiste em selecior k subconjuntos distintos S1, S2, ... , Sk em S cujo tamanho da interseção de |S1 ? S2 ? ... ? Sk| seja máxima. Trata-se de um problema NP-difícil e difícil de ser aproximado que ocorre em aplicações de áreas como biologia computacional e privacidade de dados. Até o nosso conhecimento, nenhum método exato foi proposto para resolver este problema. Neste trabalho, introduzimos cinco formulações de programação linear inteira para o problema, sendo três baseadas no método de Branch-and-Bound e duas no método de Branch-and-Cut. Além disso, uma heurística gulosa e uma meta-heurística GRASP foram desenvolvidas com o intuito de gerar bons limitantes inferiores. A heurística GRASP desenvolvida foi capaz de encontrar soluções muito próximas da solução ótima. Ademais, introduzimos um método muito eficiente de pré-processamento para reduzir o tamanho da entrada. Experimentos computacionais foram realizados de forma a analisar o desempenho dos modelos de programação linear inteira em questão, demonstrando que os modelos baseados no método de Branch-and-Cut obtiveram melhores resultados
Abstract: In this project, we study the Maximum k-Subset Intersection problem (kMIS). Given an integer k, a ground set U and a collection S of subsets of U, the kMIS problem is to select k distinct subsets S1, S2, ... , Sk in S whose intersection size |S1 ? S2 ? ... ? Sk| is maximum. The kMIS problem is NP-hard and hard to approximate and occurs in areas of applications such as computational biology and data privacy. To the best of our knowledge, no exact method was proposed to solve this problem. In this work, we introduce five integer linear programming formulations for the problem, three using a simple Branch-and-Bound method and two using a Branch-and-Cut method. We also present a greedy heuristic and a metaheuristic GRASP developed in order to generate good lower bounds. The heuristic GRASP developed was able to find solutions very close to the optimal ones. Furthermore, we introduce a very efficient preprocessing procedure to reduce the size of the input. Computational experiments were performed in order to analyze the performance of the integer linear programming models in question, showing that the Branch-and-Cut models performed better
Mestrado
Ciência da Computação
Mestre em Ciência da Computação
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Crepaldi, Bruno Espinosa 1991. "Um algoritmo eficiente para o problema do posicionamento natural de antenas". [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/275534.
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Orientadores: Cid Carvalho de Souza, Pedro Jussieu de RezendeDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação
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Resumo: Considerado uma variação do problema da galeria de arte, o problema do posicionamento de antenas trata do posicionamento do menor número de antenas requerido para determinar se uma pessoa está dentro ou fora da galeria. Uma antena propaga uma chave única dentro de um ângulo específico de transmissão, de modo que o conjunto de chaves recebidas em um dado ponto do plano seja suficiente para decidir se ele pertence ou não ao polígono que representa a galeria. Para verificar esta propriedade de localização, uma fórmula Booleana deve ser produzida junto com o posicionamento de antenas. Dizemos que as antenas estão em posição natural se elas estão localizadas nos vértices ou nas arestas do polígono e transmitindo sinal no ângulo formado pelos lados deste último no ponto onde a antena está posicionada. O problema do posicionamento natural de antenas é NP-difícil. Nesta dissertação, apresentamos um algoritmo exato para resolvê-lo. Para tanto, propomos um modelo inicial de programação linear inteira para o problema que, ao ser computado por um resolvedor comercial, se mostrou capaz de encontrar soluções ótimas de instâncias correspondentes a polígonos com algumas dezenas de vértices. Em seguida, através de estudos de propriedades geométricas, são introduzidas várias melhorias no modelo matemático e também na forma de computá-lo. Como consequência desta pesquisa, desenvolvemos um algoritmo iterativo baseado em programação linear inteira com o qual conseguimos solucionar o problema para instâncias consideravelmente maiores. A eficiência do nosso algoritmo é certificada por resultados experimentais que compreendem as soluções ótimas de 720 instâncias de até 1000 vértices, incluindo polígono com buracos, as quais foram calculadas em menos de seis minutos em um computador desktop padrão
Abstract: Considered a variation of the art gallery problem, the wireless localization problem deals with the placement of the smallest number of broadcasting antennas required to determine if someone is inside or outside the gallery. Each antenna propagates a unique key within a certain antenna-specific angle of broadcast, so that the set of keys received at any given point is sufficient to determine whether that point is inside or outside the polygon that represents the gallery. To ascertain this localization property, a Boolean formula must be produced along with the placement of the antennas. We say that the antennas are in natural position if they are located at the vertices or the edges of the polygon and transmitting their signals in the angle formed by the sides of the polygon at the point where the antenna is positioned. The natural wireless localization problem is NP-hard. In this dissertation, we present an exact algorithm to solve it. To this end, we propose an initial integer linear programming model for the problem that, after being computed by a commercial solver, proved to be capable of finding optimal solutions for instances corresponding to polygons with tens of vertices. Then, through studies of geometric properties, several improvements are introduced in the mathematical model and also in the way of computing it. As a result of this research, we develop an iterative algorithm based on integer linear programming with which we can solve the problem for considerably larger instances. The efficiency of our algorithm is certified by experimental results comprising the solutions of 720 instances, including polygon with holes with up to 1000 vertices, in less than six minutes on a standard desktop computer
Mestrado
Ciência da Computação
Mestre em Ciência da Computação
16
Lewi, Jeremy. "Sequential optimal design of neurophysiology experiments". Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/28201.
Texto completoResumen
Thesis (M. S.)--Biomedical Engineering, Georgia Institute of Technology, 2009.Committee Co-Chair: Butera, Robert; Committee Co-Chair: Paninski, Liam; Committee Member: Isbell, Charles; Committee Member: Rozell, Chris; Committee Member: Stanley, Garrett; Committee Member: Vidakovic, Brani.
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Kim, Bosung. "Two-stage combinatorial optimization framework for air traffic flow management under constrained capacity". Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/53500.
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Air traffic flow management is a critical component of air transport operations because at some point in time, often very frequently, one of more of the critical resources in the air transportation network has significantly reduced capacity, resulting in congestion and delay for airlines and other entities and individuals who use the network. Typically, these “bottlenecks” are noticed at a given airport or terminal area, but they also occur in en route airspace. The two-stage combinatorial optimization framework for air traffic flow management under constrained capacity that is presented in this thesis, represents a important step towards the full consideration of the combinatorial nature of air traffic flow management decision that is often ignored or dealt with via priority-based schemes. It also illustrates the similarities between two traffic flow management problems that heretofore were considered to be quite distinct.
The runway systems at major airports are highly constrained resources. From the perspective of arrivals, unnecessary delays and emissions may occur during peak periods when one or more runways at an airport are in great demand while other runways at the same airport are operating under their capacity. The primary cause of this imbalance in runway utilization is that the traffic flow into and out of the terminal areas is asymmetric (as a result of airline scheduling practices), and arrivals are typically assigned to the runway nearest the fix through which they enter the terminal areas. From the perspective of departures, delays and emissions occur because arrivals take precedence over departures with regard to the utilization of runways (despite the absence of binding safety constraints), and because arrival trajectories often include level segments that ensure “procedural separation” from arriving traffic while planes are not allowed to climb unrestricted along the most direct path to their destination. Similar to the runway systems, the terminal radar approach control facilities (TRACON) boundary fixes are also constrained resources of the terminal airspace. Because some arrival traffic from different airports merges at an arrival fix, a queue for the terminal areas generally starts to form at the arrival fix, which are caused by delays due to heavy arriving traffic streams. The arrivals must then absorb these delays by path stretching and adjusting their speed, resulting in unplanned fuel consumption. However, these delays are often not distributed evenly. As a result, some arrival fixes experience severe delays while, similar to the runway systems, the other arrival fixes might experience no delays at all. The goal of this thesis is to develop a combined optimization approach for terminal airspace flow management that assigns a TRACON boundary fix and a runway to each flight while minimizing the required fuel burn and emissions. The approach lessens the severity of terminal capacity shortage caused by and imbalance of traffic demand by shunting flights from current positions to alternate runways. This is done by considering every possible path combination. To attempt to solve the congestion of the terminal airspace at both runways and arrival fixes, this research focuses on two sequential optimizations. The fix assignments are dealt with by considering, simultaneously, the capacity constraints of fixes and runways as well as the fuel consumption and emissions of each flight. The research also develops runway assignments with runway scheduling such that the total emissions produced in the terminal area and on the airport surface are minimized.
The two-stage sequential framework is also extended to en route airspace. When en route airspace loses its capacity for any reason, e.g. severe weather condition, air traffic controllers and flight operators plan flight schedules together based on the given capacity limit, thereby maximizing en route throughput and minimizing flight operators' costs. However, the current methods have limitations due to the lacks of consideration of the combinatorial nature of air traffic flow management decision. One of the initial attempts to overcome these limitations is the Collaborative Trajectory Options Program (CTOP), which will be initiated soon by the Federal Aviation Administration (FAA). The developed two-stage combinatorial optimization framework fits this CTOP perfectly from the flight operator's perspective. The first stage is used to find an optimal slot allocation for flights under satisfying the ration by schedule (RBS) algorithm of the FAA. To solve the formulated first stage problem efficiently, two different solution methodologies, a heuristic algorithm and a modified branch and bound algorithm, are presented. Then, flights are assigned to the resulting optimized slots in the second stage so as to minimize the flight operator's costs.
18
Vignatti, André Luís. "Aproximação e compartilhamento de custos em projeto de redes". [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/276228.
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Orientador: Flavio Keidi MiyazawaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação
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Resumo: Neste trabalho estudamos a interação entre duas áreas: otimização combinatória e compartilhamento de custos (cost-sharing), que é a arte de dividir os custos associados a construção e manutenção de uma solução a qual um grupo de usuários é beneficiado. Apresentamos algoritmos para problemas de projeto de redes, tendo como objetivo principal os problemas ¿Connected Facility Location¿ e ¿Rent-or-Buy¿. Estes dois problemas são NP-difíceis, pois têm como caso particular o problema da arvore mínima de Steiner, que tambem é NP-dificil. Na primeira parte do trabalho, temos a seguinte questão como motivação: ¿Como projetar uma boa rede, ou seja, uma rede que satisfaça todas as propriedades do problema e ao mesmo tempo minimize o custo de construção desta rede?¿ 'E nesta parte que os algoritmos de aproximação entram em ação. Uma vez que esse custo for determinado, na segunda parte do trabalho, uma outra questão surge: ¿Como dividir esse custo entre todos os usuários que participam da rede de uma maneira ¿justa¿? Nesta parte, usaremos o compartilhamento de custos juntamente com as tecnicas de algoritmos de aproximação para responder a essa questão
Abstract: We consider the interplay of two areas: combinatorial optimization and cost-sharing in network design problems. In the first, we are interested to find a solution with small cost. In the second we would like to share the solution cost between its users. We present algorithms for the problems ¿Connected Facility Location¿ and ¿Rent-or-Buy¿. These two problems are NP-hard, since they have as a particular case the minimum Steiner tree problem, which is a known NP-hard problem. In the first part of this work, we have the following question as motivation: ¿how to design a good network, i.e., one that satisfies all problem requirements and minimize the overall network construction cost?¿ In this part, approximation algorithms takes action. Once this cost is determinated, in the second part of the work, another question arises: ¿How to distribute this cost among all users that participate in the network in a ¿fair¿ way? In this part, we will use cost-sharing together with approximation algorithms techniques to answer this question
Mestrado
Teoria da Computação
Mestre em Ciência da Computação
19
França, Fabricio Olivetti de. "Algoritmos bio-inspirados aplicados a otimização dinamica". [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/259091.
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Orientadores: Fernando Jose Von Zuben, Leandro Nunes de CastroDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Resumo: Esta dissertação propõe algoritmos bio-inspirados para a solução de problemas de otimização dinâmica, ou seja, problemas em que a superfície de otimização no espaço de busca sofre variações diversas ao longo do tempo. Com a variação, no tempo, de número, posição e qualidade dos ótimos locais, as técnicas de programação matemática tendem a apresentar uma acentuada degradação de desempenho, pois geralmente foram concebidas para tratar do caso estático. Algoritmos populacionais, controle dinâmico do número de indivíduos na população, estratégias de busca local e uso eficaz de memória são requisitos desejados para o sucesso da otimização dinâmica, sendo contemplados nas propostas de solução implementadas nesta dissertação. Os algoritmos a serem apresentados e comparados com alternativas competitivas presentes na literatura são baseados em funcionalidades e estruturas de processamento de sistemas imunológicos e de colônias de formigas. Pelo fato de considerarem todos os requisitos para uma busca eficaz em ambientes dinâmicos, o desempenho dos algoritmos imuno-inspirados se mostrou superior em todos os critérios considerados para comparação dos resultados dos experimentos.
Abstract: This dissertation proposes bio-inspired algorithms to solve dynamic optimization problems, i.e., problems for which the optimization surface on the search space suffers several changes over time. With such variation of number, position and quality of local optima, mathematical programming techniques may present degradation of performance, because they were usually conceived to deal with static problems. Population-based algorithms, dynamic control of the population size, local search strategies and an efficient memory usage are desirable requirements to a proper treatment of dynamic optimization problems, thus being incorporated into the solution strategies implemented here. The algorithms to be presented, and compared with competitive alternatives available in the literature, are based on functionalities and processing structures of immune systems and ant colonies. Due to the capability of incorporating all the requirements for an efficient search on dynamic environments, the immune-inspired approaches overcome the others in all the performance criteria adopted to evaluate the experimental results.
Mestrado
Engenharia de Computação
Mestre em Engenharia Elétrica
20
Gomes, Rafael Martins. "Otimização linear aplicada ao plantio sustentável de vegetais". Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-30082011-140705/.
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O planejamento de rotações de culturas é um tema de interesse em ascensão por permitir uma redução significativa no uso de adubos industriais, agrotóxicos e outros produtos químicos no cultivo, permitindo a auto-sustentação e qualidade das terras cultivadas. Este trabalho centraliza em utilizar rotações para atender uma demanda periódica prédeterminada, respeitando as restrições relativas a aspectos ecológicos que auxiliam na estabilidade geral do solo para definir uma rotação de culturas factível. Modelos matemáticos que consideram um tamanho mínimo de lote a ser usado por uma rotação e métodos heurísticos, baseados em geração de colunas, são apresentados. Uma análise detalhada do comportamento dos métodos perante variações em diferentes parâmetros e critérios é realizada. A primeira heurística, denominada Algoritmo GC-BC, obteve resultados de melhor qualidade e de forma mais rápida que a segunda heurística, denominada Heurística Lote Fixo. Entretanto, combinando ambas heurísticas foi possível obter os resultados mais satisfatórios, ou seja, soluções que respeitam a condição de lote mínimo em um tempo computacional aceitável para um planejamento anual, cujos valores são próximos a um limitante superior. A ideia subjacente de gerar colunas adicionais para um problema mestre restrito produz soluções de qualidade, o que pode vir a ser aplicado em outras áreas de pesquisa que necessitam da geração de colunas para uma resolução em tempo computacional viávelThe crop rotation planning is a rising topic for providing a significative reduction on the usage of industrial fertilizers, pesticides and other chemical, allowing the soil to selfsustain. This study focus on using rotations to meet a periodic and pre-defined demand while ecologic restrictions, that help sustain the soils stability, define a valid crop rotation. Mathematical models that consider a minimum size of a used lot associated with a given rotation and heuristic resolution methods, based on column generation, are presented. A detailed analysis of the methods behaviour before changes on parameters and criteria is performed. The first heuristic, called GC-BC Algorithm, achieved better and faster results compared to the second heuristic, called Fixed Lot Heuristic. However, combining both heuristics produced even better results, that is, solutions that respect the minimum lot sizing restrictions in good execution time for an annual planning. The idea behind of generating additional columns to the restricted master problem produces good quality solutions, which may be applicable in other research areas that require column generation for their resolution with a reasonable execution time
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Umetani, Shunji, Mutsunori Yagiura y 睦憲 柳浦. "RELAXATION HEURISTICS FOR THE SET COVERING PROBLEM". 日本オペレーションズ・リサーチ学会, 2007. http://hdl.handle.net/2237/11724.
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PARENTE, ANGELO. "Ricerca di strutture speciali in problemi di programmazione lineare intera". Doctoral thesis, Università Politecnica delle Marche, 2013. http://hdl.handle.net/11566/242727.
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I problemi di Programmazione Lineare Intera sono in generale notoriamente
difficili da trattare sotto l’aspetto computazionale. La progettazione di algoritmi
efficienti per problemi di tale natura, quindi, spesso trae grande vantaggio
dallo studio della struttura matematica soggiacente. Prendendo come spunto
il problema di determinare una sottomatrice massima di tipo network di una
matrice a valori in {0,−1, 1}, nel lavoro di tesi si analizza il problema equivalente
di bilanciamento di un grafo signed (mbs, Max Balanced Subgraph)
e quindi si generalizzano alcuni dei risultati ottenuti a problemi combinatorici
di più ampia portata.
In particolare, lo studio ha portato all’ideazione e implementazione dell’euristica
CCH (Cycle-Contraction Heuristic) per il problema mbs che è risultata
più efficace delle euristiche esistenti in letteratura. L’algoritmo CCH è basato
su una regola originale di trasformazione e semplificazione dei grafi che ne riduce
progressivamente la lunghezza dei cicli senza per questo compromettere
l’ammissibilità delle soluzioni determinate sul grafo modificato.
Successivamente sono state ideate delle regole originali di data reduction che
hanno lo scopo di semplificare le istanze e/o ridurne la dimensione, pur conservando
su esse la soluzione ottima del problema. Con queste regole, le performance
dell’euristica CCH sono migliorate sia in termini di efficienza che di
efficacia.
Nella tesi viene poi proposto un nuovo modello esatto per il problema mbs
basato sulla trasformazione di un grafo signed in un grafo semplice detto 2-
layer, stabilendo una stretta connessione tra il problema mbs e il più comune
problema di massimo insieme stabile (mis). Il grafo 2-layer fornisce nuovi
strumenti per progettare algoritmi per il problema mbs, basati sui numerosi
metodi di soluzione (esatti o euristici) adottati per il problema mis.
Infine, è stato affrontato il tema della generalizzazione di alcuni problemi
combinatorici su grafi signed ed è stato esteso il modello 2-layer al modello
k-layer. In particolare è stato provato che il problema di k-coloring di un grafo
signed - che generalizza il problema di bilanciamento di un grafo signed e quello
di bipartizione di un grafo semplice - equivale al problema mis sul grafo k-layer.In general, Integer Linear Programming problems are computationally hard to solve. The design of efficient algorithms for them often takes advantage from the analysis of the problem underlying mathematical structure. Starting from the problem of finding the maximum embedded network submatrix of a matrix with entries in {0,−1, 1}, this work deals with the equivalent problem of finding the maximum balanced induced subgraph of a signed graph (mbs, Max Balanced Subgraph). In particular, a new heuristic for the mbs problem, the Cycle-Contraction Heuristic (CCH), has been proposed. The algoritm is based on a graph trans- formation rule that progressively reduces the lengths of cycles, preserving at the same time the feasibility of solutions for the mbs problem. CCH turns out to be more effective of the state-of-the-art heuristics. The efficiency and the effectiveness of CCH can be further improved by means of new rules of data reduction, i.e, by a procedure that simplifies instances and/or decrease their size while preserving the optimal solution of the problem. In the second part of the work, a new exact approach for the mbs prob- lem has been proposed. Such method is based on a polynomial complexity transformation rule that turns a signed graphs into a simple 2-layer graph. The transformation estabilishes a strong connection between mbs and the well- known maximun independent set problem (mis) and allows to resort to a broad spectrum of (exact or heuristic) solution methods proposed in the literature for mis. Finally, the generalized counterpart on signed graphs of some well-known combinatorial problems have been investigated. In particular it has been proven that the k-coloring problem on a signed graph - the generalization on signed graph of the balancing problem and the generalization on a simple graph of the bipartization problem - is equivalent to mis problem on a k-layer graph, i.e., a simple graph obtained by generalizing the 2-layer transformation.
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Porretta'S, Luciano. "MODELS AND METHODS IN GENOME WIDE ASSOCIATION STUDIES". Doctoral thesis, Universite Libre de Bruxelles, 2018. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/265314.
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The interdisciplinary field of systems biology has evolved rapidly over the last few years. Different disciplines have contributed to the development of both its experimental and theoretical branches.Although computational biology has been an increasing activity in computer science for more than a two decades, it has been only in the past few years that optimization models have been increasingly developed and analyzed by researchers whose primary background is Operations Research(OR). This dissertation aims at contributing to the field of computational biology by applying mathematical programming to certain problems in molecular biology.Specifically, we address three problems in the domain of Genome Wide Association Studies}:(i) the Pure Parsimony Haplotyping Under uncertatind Data Problem that consists in finding the minimum number of haplotypes necessary to explain a given set of genotypes containing possible reading errors; (ii) the Parsimonious Loss Of Heterozygosity Problem that consists of partitioning suspected polymorphisms from a set of individuals into a minimum number of deletion areas; (iii) and the Multiple Individuals Polymorphic Alu Insertion Recognition Problem that consists of finding the set of locations in the genome where ALU sequences are inserted in some individual(s).All three problems are NP-hard combinatorial optimization problems. Therefore, we analyse their combinatorial structure and we propose an exact approach to solution for each of them. The proposed models are efficient, accurate, compact, polynomial-sized and usable in all those cases for which the parsimony criterion is well suited for estimation.Option Informatique du Doctorat en Sciences
info:eu-repo/semantics/nonPublished
24
Krause, Jonas. "Programação matemática e evolução diferencial para a otimização de redes de dutos". Universidade Tecnológica Federal do Paraná, 2013. http://repositorio.utfpr.edu.br/jspui/handle/1/752.
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A otimização de uma rede de transporte de derivados de petróleo é um problema complexo e abordado na literatura atual. A modelagem matemática deste problema proposta neste trabalho cria um problema de otimização combinatorial. Métodos de resolução deste problema através da programação linear inteira mista e de algoritmos heurísticos de evolução diferencial (Evolução Diferencial Binária e Evolução Diferencial Discretizada) são propostos utilizando variáveis binárias. Os resultados encontrados com a programação linear apresentam valores ótimos para os benchmarks com pequenos espaços de busca e valores sub-ótimos para grandes. Resultados utilizando a evolução diferencial também são apresentados como uma alternativa de baixo esforço computacional. A aplicação destes métodos proporciona alternativas para o transporte de diferentes produtos em um horizonte de tempo definido e compara os métodos heurísticos com codificações binárias e contínuas. Tais resultados incentivam a utilização de algoritmos heurísticos com codificação contínua e apontam os métodos de discretização como alternativas eficazes para a resolução de problemas discretos.The optimization of an pipeline network is a complex problem and addressed in the current literature. The mathematical modeling of this problem proposed in this paper creates a problem of combinatorial optimization. Methods for solving this problem using linear mixed integer programming and heuristic algorithms of differential evolution (Binary Differential Evolution and Discretized Differential Evolution) are proposed using binary variables. The results obtained with the linear programming have optimal values for the benchmarks with small search spaces and sub-optimal for large values. Results using the differential evolution are also presented as an alternative low computational effort. The application of these methods provides alternatives for transporting different products in a defined time horizon and compare heuristic methods with continuous and binary encodings. Such results encourage the use of heuristic algorithms with continuous coding and the point discretization methods as effective for solving problems discrete alternatives.
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Sarpong, Boadu Mensah. "Column generation for bi-objective integer linear programs : application to bi-objective vehicle routing problems". Phd thesis, INSA de Toulouse, 2013. http://tel.archives-ouvertes.fr/tel-00919861.
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L'optimisation multi-objectif concerne la résolution de problèmes pour lesquels plusieurs objectifs (ou critères) contradictoires sont pris en compte. Contrairement aux problèmes d'optimisation ayant un seul objectif, un problème multi-objectif ne possède pas une valeur optimale unique mais plutôt un ensemble de points appelés "ensemble non dominé". Les bornes inférieures et supérieures d'un problème multi-objectif peuvent être également décrites par des ensembles. Dans la pratique, les variables utilisées en optimisation multiobjectif représentent souvent des objets non fractionnables et on parle alors de problèmes multi-objectif en nombres entiers. Afin d'obtenir de meilleures bornes qui peuvent être utilisées dans la conception de méthodes exactes, certains problèmes sont formulés avec un nombre exponentiel de variables de décision et ces problèmes sont résolus par la méthode de génération de colonnes. Les travaux de cette thèse visent à contribuer à l'étude de l'utilisation de la génération de colonnes en programmation linéaires en nombres entiers multi-objectif. Pour cela nous étudions un problème de tournées de véhicules bi-objectif qui peut être considéré comme une généralisation de plusieurs autres problèmes de tournées de véhicules. Nous proposons des formulations mathématiques pour ce problème et des techniques pour accélérer le calcul des bornes inférieures par génération de colonnes. Les sous-problèmes qui doivent être résolus pour le calcul des bornes inférieures ont une structure similaire. Nous exploitons cette caractéristique pour traiter simultanément certains sous-problèmes plutôt qu'indépendamment.
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Morin, Pierre-Antoine. "Planification et ordonnancement de projets sous contraintes de ressources complexes". Thesis, Toulouse 3, 2018. http://www.theses.fr/2018TOU30291/document.
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La structure de projet se retrouve dans de nombreux contextes de l'industrie et des services. Il s'agit de réaliser un ensemble d'activités pouvant être connectées par des liens logiques de séquence (antériorité), en faisant appel à des ressources disponibles en quantité limitée. L'objectif est la minimisation d'un critère généralement lié à la durée ou au coût du projet. La plupart des problèmes d'ordonnancement de projet dans la littérature considèrent une unité de temps commune pour la détermination des dates d'exécution des activités et pour l'évaluation instantanée du respect des capacités des ressources qu'elles utilisent. Or, s'il est souvent nécessaire en pratique d'obtenir un calendrier détaillé des plages d'exécution des activités, l'utilisation des ressources peut être évaluée sur un horizon plus agrégé, comme par exemple les quarts de travail des employés. Dans cette thèse, un nouveau modèle intégrant ces deux échelles de temps est présenté afin de définir le problème d'ordonnancement de projet avec agrégation périodique des contraintes de ressources (PARCPSP). Ce problème est étudié du point de vue de la théorie de la complexité et des propriétés structurelles sont établies, mettant notamment en évidence des différences majeures avec le problème classique d'ordonnancement de projet sous contraintes de ressources (RCPSP). De ces propriétés sont dérivées des formulations exactes basées sur la programmation linéaire en nombres entiers, comparées en termes de qualité de la relaxation linéaire. Par ailleurs, plusieurs heuristiques, telles que des algorithmes de liste, ou une méthode approchée basée sur une résolution itérative qui exploite différentes échelles de temps, sont proposées. Les résultats expérimentaux montrent l'intérêt de ces différentes méthodes et illustrent la difficulté du problèmeThe project structure arises in many fields of industry and services. It consists in performing a set of activities that may be linked by precedence relations, and use resources whose capacity is limited. The objective is to minimize a criterion usually linked to the duration or the cost of the project. Most of project scheduling problems in the literature assume that the same time scale should be used to determine activity start and completion dates and check resource constraints at each time. However, although it is often required in practice to build a precise schedule specifying the execution range of each activity, the resource usage can be evaluated on an aggregated basis, like worker shifts. In this thesis, a new model that enables the integration of these two time scales is presented in order to define the periodically aggregated resource-constrained project scheduling problem (PARCPSP). This problem is studied within the framework of complexity theory and several structural properties are established, highlighting major differences with the standard resource-constrained project scheduling problem (RCPSP). These properties allow deriving exact formulations based on integer linear programming, whose linear relaxations are compared. Moreover, several heuristics, such as schedule generations schemes, or an approached method based on a multi time scale iterative process, are proposed. Experimental results show the interest of these different methods and point out the intractability of the problem
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Magatão, Leandro. "Mixed integer linear programming and constraint logic programming : towards a unified modeling framework". Centro Federal de Educação Tecnológica do Paraná, 2005. http://repositorio.utfpr.edu.br/jspui/handle/1/86.
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The struggle to model and solve Combinatorial Optimization Problems (COPs) has challenged the development of new approaches to deal with COPs. In one of the front lines of such approaches, Operational Research (OR) and Constraint Programming (CP) optimization techniques are beginning to converge, despite their very different origins. More specifically, Mixed Integer Linear Programming (MILP) and Constraint Logic Programming (CLP) are at the confluence of the OR and the CP fields. This thesis summarizes and contrasts the essential characteristics of MILP and CLP, and the ways that they can be fruitfully combined. Chapters 1 to 3 sketch the intellectual background for recent efforts at integration and the main results achieved. In addition, these chapters highlight that CLP is known by its reach modeling framework, and the MILP modeling vocabulary is just based on inequalities, which makes the modeling process hard and error-prone. Therefore, a combined CLP-MILP approach suffers from this MILP inherited drawback. In chapter 4, this issue is addressed, and some "high-level" MILP modeling structures based on logical inference paradigms are proposed. These structures help the formulation of MILP models, and can be seen as a contribution towards a unifying modeling framework for a combined CLP-MILP approach. In addition, chapter 5 presents an MILP formulation addressing a combinatorial problem. This problem is focused on issues regarding the oil industry, more specifically, issues involving the scheduling of operational activities in a multi-product pipeline. Chapter 5 demonstrates the applicability of the high-level MILP modeling structures in a real-world scenario. Furthermore, chapter 6 presents a CLP-MILP formulation addressing the same scheduling problem previously exploited. This chapter demonstrates the applicability of the high-level MILP modeling structures in an integrated CLP-MILP modeling framework. The set of simulations conducted indicates that the combined CLP-MILP model was solved to optimality faster than either the MILP model or the CLP model. Thus, the CLP-MILP framework is a promising alternative to deal with the computational burden of this pipeline-scheduling problem. In essence, this thesis considers the integration of CLP and MILP in a modeling standpoint: it conveys the fundamentals of both techniques and the modeling features that help establish a combined CLP-MILP approach. Herein, the concentration is on the building of MILP and CLP-MILP models rather than on the solution process.
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Bitar, Abdoul. "Ordonnancement sur machines parallèles appliqué à la fabrication de semi-conducteurs : ateliers de photolithographie". Thesis, Saint-Etienne, EMSE, 2015. http://www.theses.fr/2015EMSE0808/document.
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Le secteur des semi-conducteurs a connu un développement considérable ces dernières décennies, du fait des nouvelles applications de la microélectronique dans l'industrie. Le processus de fabrication est réputé pour sa complexité. L'un des ateliers les plus critiques de la production, l'atelier de photolithographie, est régi par un ensemble conséquent de contraintes de production. La multiplicité des ressources utilisées, le nombre important de produits traités, en font une zone importante à optimiser. Les objectifs de la thèse ont été de modéliser cet atelier sous la forme d'un problème d'ordonnancement sur machines parallèles et d'optimiser plusieurs critères jugés pertinents pour évaluer la qualité des solutions. Des résultats en termes de complexité, et d'algorithmes de résolution, ont permis une application industrielle, dans la mesure où un logiciel d'optimisation destiné à l'ordonnancement des lots en photolithographie a été développéSemiconductor manufacturing has grown considerably in recent decades, due to new industrial applications of microelectronic devices. The related manufacturing process is known to be complex. A bottleneck process step, the photolithography workshop, gathers various types of constraints, related to the number of auxiliary resources and the tools characteristics. The aims of the thesis were to model this workstation as a parallel machine scheduling problem and to optimize various criteria, determined by industrial needs. Some complexity results are provided and optimization algorithms led to an industrial application, i.e. a software providing optimized schedules in a specific fab
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Rodrigues, Raildo Barros. "Modelo de programação matemática na elaboração de quadros de horários para cursos de graduação". Universidade Estadual Paulista (UNESP), 2018. http://hdl.handle.net/11449/157117.
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Esta dissertação trata da construção de um modelo matemático para a elaboração do quadro de horários dos cursos de graduação do CBV/IFRR. A programação de horários é um problema de otimização combinatória estudado há anos pela Pesquisa Operacional e, em termos de complexidade computacional, é tido como NP-Completo, sendo assim, é um problema que exige grande capacidade de processamento. A elaboração do quadro de horários em qualquer instituição de ensino é complexa e demanda tempo para os responsáveis por essa atividade, pois as necessidades dos professores e alunos devem ser atendidas e devem-se evitar conflitos nos horários dos professores. A instituição estudada nesta dissertação assim como outras instituições, possui particularidades institucionais, dessa forma, uma formulação geral do problema acaba não lhe sendo útil. O CBV/IFRR realiza a elaboração dos horários de forma manual, por meio de planilha eletrônica e realização de reuniões entre os gestores, o que torna difícil encontrar uma solução factível. Sendo assim, foi necessária a realização de pesquisa científica para encontrar métodos que poderiam ser aplicados ao problema. Assim, este trabalho teve como objetivo desenvolver um modelo de Programação Matemática que permitisse a elaboração dos horários para cursos de graduação do CBV/IFRR. Utilizou-se entrevistas com as Coordenações de Cursos para obtenção das informações acerca do problema tratado, tais como restrições e prioridades a serem atendidas com a programação de aulas para professores. Estas informações serviram de base para a construção do modelo conceitual, que foi utilizado para elaboração do modelo matemático final, que foi implementado na linguagem de alto nível GAMS® e resolvido pelo solver CPLEX®. Os testes do modelo foram realizados otimizando uma instância com dados reais da instituição estudada. Os resultados obtidos da otimização foram satisfatórios, pois foi possível encontrar uma solução ótima para a instância em tempo computacional adequado, com todas as restrições, impostas pelas características peculiares do problema tratado, sendo respeitadas e as prioridades estabelecidas pelas Coordenações de Cursos atendidas.
This dissertation deals with the construction of a mathematical model for the elaboration of the timetable of the undergraduate courses of the CBV/IFRR. Time scheduling is a combinatorial optimization problem that has been studied for years by Operational Research and, in terms of computational complexity, is considered as NP-Complete, so it is a problem that requires large processing capacity. The elaboration of the timetable in any educational institution is complex and takes time for those responsible for this activity, because the needs of teachers and students must be met and avoid conflicts in the schedules of teachers. The institution studied in this dissertation as well as other institutions, has institutional features, so a general formulation of the problem ends up being of no use to it. The CBV/IFRR performs the elaboration of the schedules manually, through a spreadsheet and holding meetings between managers, which makes it difficult to find a feasible solution. Thus, it was necessary to carry out scientific research to find methods that could be applied to the problem. Thus, this work had the objective of developing a Mathematical Programming model that allowed the elaboration of the schedules for the undergraduate courses of the CBV/IFRR. We used interviews with the Course Coordinators to obtain information about the problem, such as constraints and priorities to be met with the programming of classes for teachers. This information was the basis for the construction of the conceptual model, which was used to elaborate the final mathematical model, which was implemented in the GAMS® high-level language and solved by the CPLEX® solver. The tests of the model were performed optimizing an instance with real data of the studied institution. The results obtained from the optimization were satisfactory, since it was possible to find an optimal solution for the instance in adequate computational time, with all the restrictions imposed by the peculiar characteristics of the problem, being respected and the priorities established by the Coordination of Courses attended.
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Mancel, Catherine. "Modelisation et resolution de problemes d'optimisation combinatoire issus d'applications spatiales". Phd thesis, INSA de Toulouse, 2004. http://tel.archives-ouvertes.fr/tel-00009238.
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Nos travaux portent sur la modelisation et la resolution de problemes d'optimisation combinatoire emergeant dans le cadre de la planification de missions spatiales. Ces problemes de grande taille presentent des caracteristiques communes en termes de types de donnees, de contraintes et de criteres a optimiser. Nous nous focalisons sur l'apport de la programmation lineaire pour ces problemes, associee a des methodes de simplification de l'espace de recherche, par decomposition ou grace a des techniques de propagation de contraintes. Nous avons plus particulierement etudie deux problemes. Le premier concerne la planification de communications sonde/satellite et d'experiences dans un projet d'exploration martienne. Une decomposition de ce probleme permet de le formuler comme deux problemes independants : un probleme de planification des communications que nous modelisons et resolvons par programmation lineaire en nombres entiers, et un probleme d'aide a la decision pour la planification des experiences, pour lequel nous etablissons des courbes d'evaluation de la charge des ressources deduites de l'application de techniques de propagation de contraintes basees sur un raisonnement energetique. Le second probleme etudie est celui de la planification de prises de vue d'un satellite d'observation de la Terre. Nous proposons un modele lineaire en variables mixtes et nous developpons une approche de resolution par generation de colonnes, qui est une adaptation de la programmation lineaire au traitement de problemes de grande taille, faisant appel a certaines techniques de decomposition des modeles.
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Abdallah, Fadel. "Optimization and Scheduling on Heterogeneous CPU/FPGA Architecture with Communication Delays". Thesis, Université de Lorraine, 2017. http://www.theses.fr/2017LORR0301.
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Le domaine de l'embarqué connaît depuis quelques années un essor important avec le développement d'applications de plus en plus exigeantes en calcul auxquels les architectures traditionnelles à base de processeurs (mono/multi cœur) ne peuvent pas toujours répondre en termes de performances. Si les architectures multiprocesseurs ou multi cœurs sont aujourd'hui généralisées, il est souvent nécessaire de leur adjoindre des circuits de traitement dédiés, reposant notamment sur des circuits reconfigurables, permettant de répondre à des besoins spécifiques et à des contraintes fortes particulièrement lorsqu'un traitement temps-réel est requis. Ce travail présente l'étude des problèmes d'ordonnancement dans les architectures hétérogènes reconfigurables basées sur des processeurs généraux (CPUs) et des circuits programmables (FPGAs). L'objectif principal est d'exécuter une application présentée sous la forme d'un graphe de précédence sur une architecture hétérogène CPU/FPGA, afin de minimiser le critère de temps d'exécution total ou makespan (Cmax). Dans cette thèse, nous avons considéré deux cas d'étude : un cas d'ordonnancement qui tient compte des délais d'intercommunication entre les unités de calcul CPU et FPGA, pouvant exécuter une seule tâche à la fois, et un autre cas prenant en compte le parallélisme dans le FPGA, qui peut exécuter plusieurs tâches en parallèle tout en respectant la contrainte surfacique. Dans un premier temps, pour le premier cas d'étude, nous proposons deux nouvelles approches d'optimisation, GAA (Genetic Algorithm Approach) et MGAA (Modified Genetic Algorithm Approach), basées sur des algorithmes génétiques. Nous proposons également de tester un algorithme par séparation et évaluation (méthode Branch & Bound). Les approches GAA et MGAA proposées offrent un très bon compromis entre la qualité des solutions obtenues (critère d'optimisation de makespan) et le temps de calcul nécessaire à leur obtention pour résoudre des problèmes à grande échelle, en comparant à la méthode par séparation et évaluation (Branch & Bound) proposée et l'autre méthode exacte proposée dans la littérature. Dans un second temps, pour le second cas d'étude, nous avons proposé et implémenté une méthode basée sur les algorithmes génétiques pour résoudre le problème du partitionnement temporel dans un circuit FPGA en utilisant la reconfiguration dynamique. Cette méthode fournit de bonnes solutions avec des temps de calcul raisonnables. Nous avons ensuite amélioré notre précédente approche MGAA afin d'obtenir une nouvelle approche intitulée MGA (Multithreaded Genetic Algorithm), permettent d'apporter des solutions au problème de partitionnement. De plus, nous avons également proposé un algorithme basé sur le recuit simulé, appelé MSA (Multithreaded Simulated Annealing). Ces deux approches proposées, basées sur les méthodes métaheuristiques, permettent de fournir des solutions approchées dans un intervalle de temps très raisonnable aux problèmes d'ordonnancement et de partitionnement sur système de calcul hétérogèneThe domain of the embedded systems becomes more and more attractive in recent years with the development of increasing computationally demanding applications to which the traditional processor-based architectures (either single or multi-core) cannot always respond in terms of performance. While multiprocessor or multicore architectures have now become generalized, it is often necessary to add to them dedicated processing circuits, based in particular on reconfigurable circuits, to meet specific needs and strong constraints, especially when real-time processing is required. This work presents the study of scheduling problems into the reconfigurable heterogeneous architectures based on general processors (CPUs) and programmable circuits (FPGAs). The main objective is to run an application presented in the form of a Data Flow Graph (DFG) on a heterogeneous CPU/FPGA architecture in order to minimize the total running time or makespan criterion (Cmax). In this thesis, we have considered two case studies: a scheduling case taking into account the intercommunication delays and where the FPGA device can perform a single task at a time, and another case taking into account parallelism in the FPGA, which can perform several tasks in parallel while respecting the constraint surface. First, in the first case, we propose two new optimization approaches GAA (Genetic Algorithm Approach) and MGAA (Modified Genetic Algorithm Approach) based on genetic algorithms. We also propose to compare these algorithms to a Branch & Bound method. The proposed approaches (GAA and MGAA) offer a very good compromise between the quality of the solutions obtained (optimization makespan criterion) and the computational time required to perform large-scale problems, unlike to the proposed Branch & Bound and the other exact methods found in the literature. Second, we first implemented an updated method based on genetic algorithms to solve the temporal partitioning problem in an FPGA circuit using dynamic reconfiguration. This method provides good solutions in a reasonable running time. Then, we improved our previous MGAA approach to obtain a new approach called MGA (Multithreaded Genetic Algorithm), which allows us to provide solutions to the partitioning problem. In addition, we have also proposed an algorithm based on simulated annealing, called MSA (Multithreaded Simulated Annealing). These two proposed approaches which are based on metaheuristic methods provide approximate solutions within a reasonable time period to the scheduling and partitioning problems on a heterogeneous computing system
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Freire, Alexandre da Silva. "Correspondência inexata entre grafos". Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/45/45134/tde-16092008-133830/.
Texto completoResumen
Sejam GI = (VI ,AI) e GM = (VM,AM) dois grafos simples. Um mapeamento de GI para GM é um conjunto de associações, tal que cada vértice de VI está associado a um vértice de VM, e cada aresta de AI está associada a um par de vértices de VM. A cada possível associação é atribuído um custo. O problema de correspondência inexata entre grafos (PCIG) consiste em encontrar um mapeamento de GI para GM, tal que a soma dos custos de suas associações seja mínima. Nesta dissertação, resumimos os resultados encontrados na literatura sobre o PCIG e algumas de suas variações. Os resultados que incluímos aqui tratam sobre a questão de como formular o PCIG e algumas de suas variações, através de programação linear inteira. Provamos alguns resultados de complexidade computacional que relacionam variações do PCIG a problemas clássicos, como isomorfismo e partição de grafos. Fornecemos uma formulação através de programação linear inteira para o PCCA (uma variante do PCIG com conexidade e cobertura de arestas). Mostramos que o PCCA é NP-difícil quando os grafos de entrada são completos ou árvores (chamamos o segundo caso de PCCA para árvores). Apresentamos uma formulação linear inteira e um algoritmo - que é polinomial se o grau máximo dos vértices de VM for limitado por uma constante - para o PCCA para árvores. Mostramos um caso especia em que o PCCA para árvores pode ser resolvido em tempo polinomial. Por último, exibimos alguns resultados experimentais, inclusive com instâncias reais de uma aplicação do problema.Let GI = (VI ,AI) and GM = (VM,AM) be two simple graphs. A mapping from GI to GM is an association set, such that each vertex in VI is associated to a vertex in VM, and each edge in AI is associated to a pair of vertices of VM. A cost is defined to each possible association. The inexact graph correspondence problem (IGCP) consists in finding a mapping from GI to GM, such that the sum of its associations costs is minimized. In this dissertation, we summarize the results found in the literature about the IGCP and some variations. The results included here address the question of how to formulate the IGCP and some variations, using integer linear programming. We prove some computational complexity results which relate IGCP variations with classical problems, like graph isomorphism and partitioning. We give an integer linear programming formulation to the ICEC (IGCP with connectivity and edges cover). We show that the ICEC is NP-hard when the input graphs are complete or trees (we call the second case ICEC for trees). We introduce an integer linear formulation and an algorithm - which has polynomial running time if the vertices of VM have maximum degree bounded by a constant - to the ICEC for trees. We show a especial case in which the ICEC for trees can be solved in polynomial time. Finally, we present some experimental results, also with instances of a real application of the problem.
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Pucu, Paulo Aliberto Barros. "Logística do escoamento da produção de petróleo de plataformas offshore via transporte naval". Universidade Federal de Alagoas, 2011. http://www.repositorio.ufal.br/handle/riufal/1280.
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Currently, Brazil has 113 petroleum platforms, been 79 fixed and 34 floating, with daily production capacity of 2,1 million barrels of oil. Given this production is necessary a strategy for the efficient distribution of oil to refineries, where it will be processed and refined. Oil from the platforms is transported to refineries through pipelines or ships, with much of the operational cost of production is due to transport. For this reason the minimization of the cost of transport is extremely important. This work has for objective, using the technique of mathematical programming (linear mixed integer programming - LMIP), reduce costs arising from transport system. The model consists of a heterogeneous fleet of ships, which have compartments that can only be occupied by a single type of product on each trip. Initially are generated all possible routes and then selected the vessels, associated with their routes in order to attend the demand of refineries and the need for removal of oil in the storage tanks of the platforms. For the implementation of the model was used the software GAMS (General Algebraic Modeling System), together with the method of CPLEX optimization. The results were satisfactory.Atualmente, o Brasil possui 113 plataformas de petróleo, sendo 79 fixas e 34 flutuantes, com capacidade de produção de 2,1 milhões de barris diários de petróleo. Diante desta produção torna-se necessária uma estratégia eficiente para a distribuição deste petróleo para as refinarias, onde será processado e refinado. O petróleo proveniente das plataformas é transportado para as refinarias, através de navios ou dutos, sendo que grande parte do custo operacional de produção é devido ao seu transporte. Por este motivo a minimização do custo de transporte é extremamente importante. Este trabalho tem por objetivo, utilizando a técnica de programação matemática (programação linear inteira mista – PLIM), reduzir os custos decorrentes do sistema de transporte. O modelo consiste em uma frota heterogênea de navios, os quais apresentam compartimentos que só podem ser ocupados por um único tipo de produto, em cada viagem. Inicialmente são geradas todas as possíveis rotas e, posteriormente, selecionados os navios, associados às respectivas rotas, de forma a atender a demanda das refinarias e a necessidade de retirada de petróleo dos tanques de armazenamento das plataformas. Para a implementação do modelo foi utilizado o software GAMS (General Algebraic Modeling System), juntamente com o método de otimização CPLEX. Os resultados obtidos foram satisfatórios.
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Abdallah, Fadel. "Optimization and Scheduling on Heterogeneous CPU/FPGA Architecture with Communication Delays". Electronic Thesis or Diss., Université de Lorraine, 2017. http://www.theses.fr/2017LORR0301.
Texto completoResumen
Le domaine de l'embarqué connaît depuis quelques années un essor important avec le développement d'applications de plus en plus exigeantes en calcul auxquels les architectures traditionnelles à base de processeurs (mono/multi cœur) ne peuvent pas toujours répondre en termes de performances. Si les architectures multiprocesseurs ou multi cœurs sont aujourd'hui généralisées, il est souvent nécessaire de leur adjoindre des circuits de traitement dédiés, reposant notamment sur des circuits reconfigurables, permettant de répondre à des besoins spécifiques et à des contraintes fortes particulièrement lorsqu'un traitement temps-réel est requis. Ce travail présente l'étude des problèmes d'ordonnancement dans les architectures hétérogènes reconfigurables basées sur des processeurs généraux (CPUs) et des circuits programmables (FPGAs). L'objectif principal est d'exécuter une application présentée sous la forme d'un graphe de précédence sur une architecture hétérogène CPU/FPGA, afin de minimiser le critère de temps d'exécution total ou makespan (Cmax). Dans cette thèse, nous avons considéré deux cas d'étude : un cas d'ordonnancement qui tient compte des délais d'intercommunication entre les unités de calcul CPU et FPGA, pouvant exécuter une seule tâche à la fois, et un autre cas prenant en compte le parallélisme dans le FPGA, qui peut exécuter plusieurs tâches en parallèle tout en respectant la contrainte surfacique. Dans un premier temps, pour le premier cas d'étude, nous proposons deux nouvelles approches d'optimisation, GAA (Genetic Algorithm Approach) et MGAA (Modified Genetic Algorithm Approach), basées sur des algorithmes génétiques. Nous proposons également de tester un algorithme par séparation et évaluation (méthode Branch & Bound). Les approches GAA et MGAA proposées offrent un très bon compromis entre la qualité des solutions obtenues (critère d'optimisation de makespan) et le temps de calcul nécessaire à leur obtention pour résoudre des problèmes à grande échelle, en comparant à la méthode par séparation et évaluation (Branch & Bound) proposée et l'autre méthode exacte proposée dans la littérature. Dans un second temps, pour le second cas d'étude, nous avons proposé et implémenté une méthode basée sur les algorithmes génétiques pour résoudre le problème du partitionnement temporel dans un circuit FPGA en utilisant la reconfiguration dynamique. Cette méthode fournit de bonnes solutions avec des temps de calcul raisonnables. Nous avons ensuite amélioré notre précédente approche MGAA afin d'obtenir une nouvelle approche intitulée MGA (Multithreaded Genetic Algorithm), permettent d'apporter des solutions au problème de partitionnement. De plus, nous avons également proposé un algorithme basé sur le recuit simulé, appelé MSA (Multithreaded Simulated Annealing). Ces deux approches proposées, basées sur les méthodes métaheuristiques, permettent de fournir des solutions approchées dans un intervalle de temps très raisonnable aux problèmes d'ordonnancement et de partitionnement sur système de calcul hétérogèneThe domain of the embedded systems becomes more and more attractive in recent years with the development of increasing computationally demanding applications to which the traditional processor-based architectures (either single or multi-core) cannot always respond in terms of performance. While multiprocessor or multicore architectures have now become generalized, it is often necessary to add to them dedicated processing circuits, based in particular on reconfigurable circuits, to meet specific needs and strong constraints, especially when real-time processing is required. This work presents the study of scheduling problems into the reconfigurable heterogeneous architectures based on general processors (CPUs) and programmable circuits (FPGAs). The main objective is to run an application presented in the form of a Data Flow Graph (DFG) on a heterogeneous CPU/FPGA architecture in order to minimize the total running time or makespan criterion (Cmax). In this thesis, we have considered two case studies: a scheduling case taking into account the intercommunication delays and where the FPGA device can perform a single task at a time, and another case taking into account parallelism in the FPGA, which can perform several tasks in parallel while respecting the constraint surface. First, in the first case, we propose two new optimization approaches GAA (Genetic Algorithm Approach) and MGAA (Modified Genetic Algorithm Approach) based on genetic algorithms. We also propose to compare these algorithms to a Branch & Bound method. The proposed approaches (GAA and MGAA) offer a very good compromise between the quality of the solutions obtained (optimization makespan criterion) and the computational time required to perform large-scale problems, unlike to the proposed Branch & Bound and the other exact methods found in the literature. Second, we first implemented an updated method based on genetic algorithms to solve the temporal partitioning problem in an FPGA circuit using dynamic reconfiguration. This method provides good solutions in a reasonable running time. Then, we improved our previous MGAA approach to obtain a new approach called MGA (Multithreaded Genetic Algorithm), which allows us to provide solutions to the partitioning problem. In addition, we have also proposed an algorithm based on simulated annealing, called MSA (Multithreaded Simulated Annealing). These two proposed approaches which are based on metaheuristic methods provide approximate solutions within a reasonable time period to the scheduling and partitioning problems on a heterogeneous computing system
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Amorim, Rainer Xavier de. "Um estudo sobre formulações matemáticas e estratégias algorítmicas para problemas de escalonamento em máquinas paralelas com penalidades de antecipação e atraso". Universidade Federal do Amazonas, 2013. http://tede.ufam.edu.br/handle/tede/2898.
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This dissertation presents a study on scheduling problems with earliness and tardiness penalties on identical parallel machines, considering independent and weighted jobs with arbitrary processing times. An analysis of the major mathematical formulations in integer programming is given, and presented the main results from the literature. An integer mathematical formulation based on network flow model was also proposed for the problem, which can be applied on single and parallel machines without idle time. Exact methods of implicit enumeration were studied and applied for the problem through the integer linear programming solver CPLEX and the UFFLP library and, mainly, algorithmic strategies of global optimization based on local search heuristic and path-relinking technique were developed. The computational experiments shows that the proposed algorithmic strategies are competitive in relation to existing results from the literature for single-machine scheduling, involving instances based on OR-Library benchmark for 40, 50, 100, 150, 200 and 300 jobs, where all the optimal values were found, and, mainly, being the best algorithmic strategy for multiprocessor environments, involving 2, 4 and 10 identical parallel machines.
Esta dissertação apresenta um estudo sobre problemas de escalonamento com penalidades de antecipação e atraso em máquinas paralelas, considerando tarefas independentes, ponderadas e de tempos de execução arbitrários. Uma análise sobre as principais formulações matemáticas em programação inteira é dada, bem como apresentados os principais resultados da literatura. Uma formulação matemática de programação inteira baseada no modelo de fluxo em redes também foi proposta para o problema, que pode ser aplicada em ambientes mono e multiprocessado sem tempo ocioso. Métodos de enumeração implícita foram estudados e aplicados aos problemas em questão através do resolvedor de programação linear inteira CPLEX e da biblioteca UFFLP, principalmente, estratégias algorítmicas aproximadas de otimização global baseadas em heurísticas de busca local e técnica de reconexão de caminhos foram desenvolvidas. Os experimentos computacionais mostram que as estratégias propostas são competitivas em relação aos resultados existentes na literatura para ambientes de escalonamento monoprocessados, envolvendo instâncias baseadas no benchmark da OR-Library para 40, 50, 100, 150, 200 e 300 tarefas, onde todos os ótimos foram encontrados, e, principalmente, sendo a melhor estratégia apresentada para ambientes multiprocessados, envolvendo 2, 4 e 10 máquinas paralelas idênticas.
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Falq, Anne-Elisabeth. "Dominances en programmation linéaire : ordonnancement autour d’une date d’échéance commune". Electronic Thesis or Diss., Sorbonne université, 2020. http://www.theses.fr/2020SORUS128.
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Les problèmes d’ordonnancement sont des problèmes d’optimisation combinatoire modélisant la gestion de projets: il s’agit de planifier l’exécution de tâches, sous des contraintes de ressources ou de précédence et de manière à minimiser un coût ou maximiser un gain. On appelle programmation linéaire en nombres entiers (PLNE) l’optimisation d’une fonction linéaire sur les points entiers vérifiant un lot de contraintes linéaires. Cet outil permet de modéliser de nombreux problèmes de recherche opérationnelle, qui peuvent alors être résolus par des solveurs implémentant l’algorithme du simplexe dans un schéma de branchement et évaluation. Cette thèse porte sur l’étude d’un problème d’ordonnancement où les tâches doivent être exécutées sur une machine de manière à minimiser les pénalités d’avance et de retard par rapport à une date de fin souhaitée commune. Grâce à des propriétés dites de dominance utilisées par la communauté de l’ordonnancement, nous avons fourni plusieurs formulations PLNE modélisant ce problème. Les premières formulations, basées sur des variables continues comparables à des dates de fin, dites variables naturelles, utilisent des inégalités de non-chevauchement. Les dernières formulations, basées sur des variables booléennes de partitions, reposent sur un type nouveau d’inégalités linéaires qui traduisant des propriétés de dominanceScheduling problems are combinatorial optimization problems arising in project management: the aim is to schedule tasks execution under resource constraints or precedence constraints so as to minimize a cost or maximize a gain. An integer linear programm (ILP° consists in optimizing a linear objective function over the integer points satisfying linear constraints. A lot of operation research problems can be formulated as ILP, and then be solved by commercial ILP. This thesis focuses on a single machine scheduling problem where earliness and tardiness with respect to a common due date have to be minimized. Thanks to so-called dominance properties used in the scheduling field, we propose several ILP formulation for this problem. First formulations, which are based on continuous variables similar to completion times variables (natural variables), use non-overlapping inequalities. Last formulations, which are based on binary partition variables, rely on a new type of linear inequalities that translate dominance properties
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PUCU, Paulo Aliberto Barros. "Desenvolvimento de um modelo matemático para minimização do custo total da operação de transporte de petróleo via marítima". Universidade Federal de Campina Grande, 2015. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/440.
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O Brasil possui atualmente 115 plataformas de petróleo, sendo 79 fixas e 34 flutuantes, com capacidade de produção de 2,1 milhões de barris diários de petróleo. Diante desta produção torna-se necessária uma estratégia eficiente para a distribuição deste petróleo para as refinarias, onde será processado e refinado. O petróleo proveniente das plataformas é transportado para as refinarias através de navios ou dutos, sendo que grande parte do custo operacional de produção é devido ao seu transporte. Por este motivo a minimização do custo de transporte é extremamente importante. Este trabalho tem por objetivo, utilizando a técnica de programação matemática (Programação Linear Inteira Mista – PLIM), reduzir os custos decorrentes do sistema de transporte. O modelo consiste em uma frota heterogênea de navios, os quais apresentam compartimentos que só podem ser ocupados por um único tipo de produto em cada viagem. Inicialmente são geradas todas as possíveis rotas e, posteriormente, selecionados os navios, associados às respectivas rotas, de forma a atender as demandas das refinarias e a necessidade de retirada de petróleo dos tanques de armazenamento das plataformas. Para a implementação do modelo foi utilizado o software GAMS (General Algebraic Modeling System), juntamente com os solveres de otimização CPLEX e BONMIN.
Currently, Brazil has 115 petroleum platforms, been 79 fixed and 34 floating, with daily production capacity of 2.1 million barrels of oil. Given this production is necessary a strategy for the efficient distribution of oil to refineries, where it will be processed and refined. Oil from the platforms is transported to refineries through pipelines or ships, with much of the operational cost of production is due to transport. For this reason the minimization of the cost of transport is extremely important. This work has for objective, using the technique of mathematical programming (linear mixed integer programming - LMIP), reduce costs arising from transport system. The model consists of a heterogeneous fleet of ships, which have compartments that can only be occupied by a single type of product on each trip. Initially are generated all possible routes and then selected the vessels, associated with their routes in order to attend the demand of refineries and the need for removal of oil in the storage tanks of the platforms. For the implementation of the model was used the software GAMS (General Algebraic Modeling System), together with the solveres of CPLEX and BONMIN optimization. The results were satisfactory.
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Hashimoto, Marcelo. "Bases de Hilbert". Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/45/45134/tde-08102007-121713/.
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Muitas relações min-max em otimização combinatória podem ser demonstradas através de total dual integralidade de sistemas lineares. O conceito algébrico de bases de Hilbert foi originalmente introduzido com o objetivo de melhor compreender a estrutura geral dos sistemas totalmente dual integrais. Resultados apresentados posteriormente mostraram que bases de Hilbert também são relevantes para a otimização combinatória em geral e para a caracterização de certas classes de objetos discretos. Entre tais resultados, foram provadas, a partir dessas bases, versões do teorema de Carathéodory para programação inteira. Nesta dissertação, estudamos aspectos estruturais e computacionais de bases de Hilbert e relações destas com programação inteira e otimização combinatória. Em particular, consideramos versões inteiras do teorema de Carathéodory e conjecturas relacionadas.There are several min-max relations in combinatorial optimization that can be proved through total dual integrality of linear systems. The algebraic concept of Hilbert basis was originally introduced with the objective of better understanding the general structure of totally dual integral systems. Some results that were proved later have shown that Hilbert basis are also relevant to combinatorial optimization in a general manner and to characterize certain classes of discrete objects. Among such results, there are versions of Carathéodory\'s theorem for integer programming that were proved through those basis. In this dissertation, we study structural and computational aspects of Hilbert basis and their relations to integer programming and combinatorial optimization. In particular, we consider integer versions of Carathéodory\'s theorem and related conjectures.
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Souza, Neto José Ferreira de. "Modelagem e meta-heurísticas para o problema de roteamento de veículos com janelas de tempo, múltiplos entregadores e múltiplas viagens em uma empresa de distribuição de bebidas". Universidade Federal de São Carlos, 2016. https://repositorio.ufscar.br/handle/ufscar/7942.
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Vehicle routing problems occur in many practical situations where the pickup and/or delivery of goods is required. In this context, the present research aims to contribute to the study of logistic operations that arise in companies that deliver products on a regular basis to customers in densely populated urban areas. The problem consists in designing minimal cost daily routes serving the maximal number of customers. To this end, the crew of each vehicle comprise multiple deliverymen as means to reduce service times. Based on a case study in a drinks producer and distributor in the state of São Paulo, it is proposed a mixed integer linear programming model that comprise costs with own and chartered vehicles and the number of deliverymen, and various operational constraints such as time windows in customers, multiple daily trips, time limitations for the circulation of some vehicle types in specific areas, compatibility between vehicles and customers, maximum load in each vehicle, maximum route time and minimum load for the realization of a second trip. Results obtained by solving the model with real instances through exact (branch&cut), heuristic (constructive, local search, GRASP and Simulated Annealing) and hybrid (GRASP and branch&cut) approaches demonstrate the good quality of the generated solutions, and indicate the potential of application of some of these methods in practice.
Problemas de roteamento de veículos ocorrem em diversas situações práticas onde se faz necessária a distribuição e/ou coleta de produtos. Nesse contexto, a presente pesquisa visa o estudo das operações logísticas presentes em empresas que entregam produtos em base regular a clientes localizados em áreas urbanas de alta densidade demográfica. O problema consiste na obtenção de rotas de mínimo custo visando o atendimento do maior número de clientes da carteira diária. Para tal, a tripulação de cada veículo pode contemplar múltiplos entregadores para redução dos tempos de serviço. Com base em um estudo de caso em uma distribuidora de bebidas do interior do Estado de São Paulo, é proposto um modelo de programação linear inteira mista que considera custos com frota própria e fretada e com o número de entregadores, e diversas restrições operacionais, tais como janelas de tempo em clientes, múltiplas viagens diárias, limitações de horários de circulação de tipos de veículos, compatibilidade entre veículos e clientes, capacidade máxima de carga a ser transportada em cada veículo, tempo máximo de rota e carga mínima para realização da segunda viagem. Resultados da resolução do modelo para instâncias reais por meio de abordagens exatas (branch&cut), heurísticas (construtiva, busca local, GRASP e Simulated Annealing) e híbrida (GRASP e branch&cut), demonstram a boa qualidade das soluções geradas, e evidenciam o potencial de uso dessas metodologias na prática.
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Bertsimas, Dimitris. "Probabilistic combinatorial optimization problems". Thesis, Massachusetts Institute of Technology, 1988. http://hdl.handle.net/1721.1/14386.
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Laksman, Efraim. "Combinatorial Optimization : Three Applications". Doctoral thesis, Blekinge Tekniska Högskola, Avdelningen för matematik och naturvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-00538.
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Combinatorial optimization is a diverse area of mathematics. It concerns optimization on feasible regions defined by discrete sets, graphs, hypergraphs, matroids, etc. . . which all have a large number of applications. They occur in virtually all domains of human activity since humans always want to do things easier, faster, consume less resources, etc. . . This thesis concerns three applied problems within combinatorial optimization. The first paper generalizes previous optimal upper bounds on the minimum Euclidean distance for phase-shift keying (PSK) block codes, that are explicit in the parameters alphabet size, block length and code size. There is a strong connection between high minimum Euclidean distance and good error-correcting capabilities. The bounds are generalized in several respects, such as from codes on symmetric PSK to codes on asymmetric PSK. They are also generalized to other types of noise than Gaussian, allowing more efficient block codes when noise is non-Gaussian. We provide examples of codes on asymmetric PSK that have higher minimum Euclidean distance than any comparable codes on symmetric PSK.Several classes of codes are shown to be optimal among codes on symmetric PSK since their Euclidean distance coincides with the bound. The second paper considers a parallel computer system with m identical computers,where we study optimal performance precaution for one possible computer crash. We anticipate that some computer may crash, and restrict the cost in such a situation. How costly is such a precaution when no crash occurs? We set a restriction that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use an optimal allocation with m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time defines a function f(r,m). In the paper we establish upper and lower bounds of the worst-case cost function f(r,m) and characterize worst-case programs. The third paper considers the problem of Map Matching (MM), i.e. matching time and location measurements of a vehicle to a route in a road network. The paper presents a probabilistic algorithm for MM based on a second order hidden Markov model (HMM), as opposed to first order HMMs which are usually used. This allows a more detailed analysis of the data while preserving algorithmic complexity O(n). Both measurement densities and transition probabilities are determined with respect to Kolmogorov's third axiom, which in this context implies that the probabilities are additive over a partition of a road segment.
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Tucker, Paul. "Two combinatorial optimization problems /". Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1999. http://wwwlib.umi.com/cr/ucsd/fullcit?p9935456.
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Skidmore, Gerald. "Metaheuristics and combinatorial optimization problems /". Online version of thesis, 2006. https://ritdml.rit.edu/dspace/handle/1850/2319.
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Roland, Julien. "Inverse multi-objective combinatorial optimization". Doctoral thesis, Universite Libre de Bruxelles, 2013. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209383.
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The initial question addressed in this thesis is how to take into account the multi-objective aspect of decision problems in inverse optimization. The most straightforward extension consists of finding a minimal adjustment of the objective functions coefficients such that a given feasible solution becomes efficient. However, there is not only a single question raised by inverse multi-objective optimization, because there is usually not a single efficient solution. The way we define inverse multi-objectiveoptimization takes into account this important aspect. This gives rise to many questions which are identified by a precise notation that highlights a large collection of inverse problems that could be investigated. In this thesis, a selection of inverse problems are presented and solved. This selection is motivated by their possible applications and the interesting theoretical questions they can rise in practice.
Doctorat en Sciences de l'ingénieur
info:eu-repo/semantics/nonPublished
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Ruzika, Stefan. "On multiple objective combinatorial optimization". München Verl. Dr. Hut, 2007. http://d-nb.info/988623870/04.
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Slusky, Marla. "Integrating Relaxations for Combinatorial Optimization". Research Showcase @ CMU, 2015. http://repository.cmu.edu/dissertations/522.
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In this thesis we explore two methods of computing lower bounds. We first discuss the Lagrangian Relaxation as it applies to the Golomb ruler problem, and then we explore adding multi-valued decision diagrams to an additive bounding scheme. The Golomb Ruler Problem asks to position n integer marks on a ruler such that all pairwise distances between the marks are distinct and the ruler has minimum total length. It is a notoriously challenging combinatorial problem, and provably optimal rulers are only known for n up to 27. Lower bounds can be obtained using linear programming (LP) formulations, but these are computationally expensive for large n. In Chapter 2 of this thesis, we propose a new method for finding lower bounds based on a Lagrangian relaxation. We apply a subgradient optimization scheme to find good bounds quickly, and we show experimentally that our method can find bounds that are very close to the optimal LP bound in a fraction of the time that is needed to compute the LP bound. We furthermore embed our Lagrangian bounds into a constraint programming search procedure, and show that these can help reduce the constraint programming search tree considerably. Additive bounding is a method of taking several algorithms for computing lower bounds, each of which typically exploits a different substructure of the problem, and combining them to produce a single lower bound which is larger than the lower bound that any of the individual algorithms can produce alone. Approximate multi-valued decision diagrams (MDDs) have recently been used to compute upper and lower bounds on several optimization problems. In Chapter 3 of this thesis, we show how we can integrate MDDs into an addivite bounding scheme.
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Udwani, Rajan. "Vignettes on robust combinatorial optimization". Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/120192.
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Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 137-142).
In this thesis, we design and analyze algorithms for robust combinatorial optimization in various settings. First, we consider the problem of simultaneously maximizing multiple objectives, all monotone submodular, subject to a cardinality constraint. We focus on the case where the number of objectives is super-constant yet much smaller than the cardinality of the chosen set. We propose several algorithms (including one with the best achievable asymptotic guarantee for the problem). Experiments on synthetic data show that a heuristic based on our more practical and fast algorithm outperforms current practical algorithms in all cases considered. Next, we study the problem of robust maximization of a single monotone submodular function in scenarios where after choosing a feasible set of elements, some elements from the chosen set are adversarially removed. Under some restriction on the number of elements that can be removed, we give the first constant factor approximation algorithms as well as the best possible asymptotic approximation in certain cases. We also give a black box result for the much more general setting of deletion-robust maximization subject to an independence system. Lastly, we consider a robust appointment scheduling problem where the goal is to design simple appointment systems that try to achieve both high server utilization as well as short waiting times, under uncertainty in job processing times. When the order of jobs is fixed and one seeks to find optimal appointment duration for jobs, we give a simple heuristic that achieves the first constant factor (2) approximation. We also give closed form optimal solutions in various special cases that supersede previous work. For the setting where order of jobs is also flexible and under-utilization costs are homogeneous, it was previously shown that an EPTAS exists. We instead focus on simple and practical heuristics, and find a ratio based ordering which is 1.0604 approximate, improving on previous results for similarly practical heuristics.
by Rajan Udwani.
Ph. D.
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Tada, Minoru. "STUDIES ON FUZZY COMBINATORIAL OPTIMIZATION". Kyoto University, 1994. http://hdl.handle.net/2433/160752.
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本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものであるKyoto University (京都大学)
0048
新制・論文博士
博士(工学)
乙第8724号
論工博第2927号
新制||工||976(附属図書館)
UT51-94-Z475
(主査)教授 茨木 俊秀, 教授 長谷川 利治, 教授 片山 徹
学位規則第4条第2項該当
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Ngulo, Uledi. "Decomposition Methods for Combinatorial Optimization". Licentiate thesis, Linköpings universitet, Tillämpad matematik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-175896.
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This thesis aims at research in the field of combinatorial optimization. Problems within this field often posses special structures allowing them to be decomposed into more easily solved subproblems, which can be exploited in solution methods. These structures appear frequently in applications. We contribute with both re-search on the development of decomposition principles and on applications. The thesis consists of an introduction and three papers. In Paper I, we develop a Lagrangian meta-heuristic principle, which is founded on a primal-dual global optimality condition for discrete and non-convex optimization problems. This condition characterizes (near-)optimal solutions in terms of near-optimality and near-complementarity measures for Lagrangian relaxed solutions. The meta-heuristic principle amounts to constructing a weighted combination of these measures, thus creating a parametric auxiliary objective function (which is a close relative to a Lagrangian function), and embedding a Lagrangian heuristic in a search procedure in the space of the weight parameters. We illustrate and assess the Lagrangian meta-heuristic principle by applying it to the generalized assignment problem and to the set covering problem. Our computational experience shows that the meta-heuristic extension of a standard Lagrangian heuristic principle can significantly improve upon the solution quality. In Paper II, we study the duality gap for set covering problems. Such problems sometimes have large duality gaps, which make them computationally challenging. The duality gap is dissected with the purpose of understanding its relationship to problem characteristics, such as problem shape and density. The means for doing this is the above-mentioned optimality condition, which is used to decompose the duality gap into terms describing near-optimality in a Lagrangian relaxation and near-complementarity in the relaxed constraints. We analyse these terms for numerous problem instances, including some large real-life instances, and conclude that when the duality gap is large, the near-complementarity term is typically large and the near-optimality term small. The large violation of complementarity is due to extensive over-coverage. Our observations have implications for the design of solution methods, especially for the design of core problems. In Paper III, we study a bi-objective covering problem stemming from a real-world application concerning the design of camera surveillance systems for large-scale outdoor areas. It is prohibitively costly to surveil the entire area, and therefore relevant to be able to present a decision-maker with trade-offs between total cost and the portion of the area that is surveilled. The problem is stated as a set covering problem with two objectives, describing cost and portion of covering constraints that are fulfilled, respectively. Finding the Pareto frontier for these objectives is very computationally demanding and we therefore develop a method for finding a good approximate frontier in a reasonable computing time. The method is based on the ε−constraint reformulation, an established heuristic for set covering problems, and subgradient optimization.Denna avhandling behandlar lösningsmetoder för stora och komplexa kombinatoriska optimeringsproblem. Sådana problem har ofta speciella strukturer som gör att de kan dekomponeras i en uppsättning mindre delproblem, vilket kan utnyttjas för konstruktion av effektiva lösningsmetoder. Avhandlingen omfattar både grundforskning inom utvecklingen av dekompositionsprinciper för kombinatorisk optimering och forskning på tillämpningar inom detta område. Avhandlingen består av en introduktion och tre artiklar. I den första artikeln utvecklar vi en “Lagrange-meta-heuristik-princip”. Principen bygger på primal-duala globala optimalitetsvillkor för diskreta och icke-konvexa optimeringsproblem. Dessa optimalitetsvillkor beskriver (när)optimala lösningar i termer av när-optimalitet och när-komplementaritet för Lagrange-relaxerade lösningar. Den meta-heuristiska principen bygger på en ihopviktning av dessa storheter vilket skapar en parametrisk hjälpmålfunktion, som har stora likheter med en Lagrange-funktion, varefter en traditionell Lagrange-heuristik används för olika värden på viktparametrarna, vilka avsöks med en meta-heuristik. Vi illustrerar och utvärderar denna meta-heuristiska princip genom att tillämpa den på det generaliserade tillordningsproblemet och övertäckningsproblemet, vilka båda är välkända och svårlösta kombinatoriska optimeringsproblem. Våra beräkningsresultat visar att denna meta-heuristiska utvidgning av en vanlig Lagrange-heuristik kan förbättra lösningskvaliteten avsevärt. I den andra artikeln studerar vi egenskaper hos övertäckningsproblem. Denna typ av optimeringsproblem har ibland stora dual-gap, vilket gör dem beräkningskrävande. Dual-gapet analyseras därför med syfte att förstå dess relation till problemegenskaper, såsom problemstorlek och täthet. Medlet för att göra detta är de ovan nämnda primal-duala globala optimalitetsvillkoren för diskreta och icke-konvexa optimeringsproblem. Dessa delar upp dual-gapet i två termer, som är när-optimalitet i en Lagrange-relaxation och när-komplementaritet i de relaxerade bivillkoren, och vi analyserar dessa termer för ett stort antal probleminstanser, däribland några storskaliga praktiska problem. Vi drar slutsatsen att när dualgapet är stort är vanligen den när-komplementära termen stor och den när-optimala termen liten. Vidare obseveras att när den när-komplementära termen är stor så beror det på en stor överflödig övertäckning. Denna förståelse för problemets inneboende egenskaper går att använda vid utformningen av lösningsmetoder för övertäckningsproblem, och speciellt för konstruktion av så kallade kärnproblem. I den tredje artikeln studeras tvåmålsproblem som uppstår vid utformningen av ett kameraövervakningssystem för stora områden utomhus. Det är i denna tillämpning alltför kostsamt att övervaka hela området och problemet modelleras därför som ett övertäckningsproblem med två mål, där ett mål beskriver totalkostnaden och ett mål beskriver hur stor del av området som övervakas. Man önskar därefter kunna skapa flera lösningar som har olika avvägningar mellan total kostnad och hur stor del av området som övervakas. Detta är dock mycket beräkningskrävande och vi utvecklar därför en metod för att hitta bra approximationer av sådana lösningar inom rimlig beräkningstid.
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Naji, Azimi Zahra <1982>. "Algorithms for Combinatorial Optimization Problems". Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2010. http://amsdottorato.unibo.it/2695/1/Naji_Azimi_Zahra_Tesi.pdf.
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