[pt] Esta tese generaliza a variante multi-período do clássico problema da Árvore de Steiner com coleta de prêmios (PCST), que visa encontrar um subgrafo conexo que maximize os prêmios recuperados de nós conectados menos o custo de utilização das arestas conectadas. Este trabalho adicionalmente: (a) permite que vértices sejam conectados à árvore em diferentes períodos de tempo; (b) impõe um orçamento pré-definido em arestas selecionadas em um horizonte específico de períodos de tempo; e (c) limita o comprimento total de arestas que podem ser adicionadas em um período de tempo. Um algoritmo branch-and-cut é fornecido para este problema, avaliando satisfatoriamente instâncias benchmark da literatura, adaptadas para uma configuração multi-período, de até aproximadamente 2000 vértices e 200 terminais em tempo razoável. / [en] This thesis generalizes the multi-period variant of the classical Prizecollecting Steiner Tree Problem, which aims at finding a connected subgraph that maximizes the revenues collected from connected nodes minus the costs to utilize the connecting edges. This work additionally: (a) allows vertices to be added to the tree at different time periods; (b) imposes a predefined budget on edges selected over a specific horizon of time periods; and (c) limits the total length of edges that can be added over a time period. A branch-and-cut algorithm is provided for this problem, satisfactorily evaluating benchmark instances from the literature, adapted to a multi-period setting, up to approximately 2000 vertices and 200 terminals in reasonable time.

Dans cette thèse, nous nous intéressons aux problèmes du knapsack multidimensionnel à choix multiple. Ils interviennent essentiellement en télécommunication. Nous proposons de nouvelles méthodes hybrides de résolution exacte et approchée. Dans un premier temps, nous proposons des méthodes heuristiques en se basant sur les techniques de génération de colonnes et d'arrondi. Ensuite, nous abordons une méthode de recherche locale, dite méthode de branchement local, où des contraintes linéaires sont introduites pour intensifier et diversifier la recherche. Cette méthode est ensuite hybridée avec la génération de colonnes et une technique d'arrondi. Concernant la résolution exacte, nous nous basons sur une méthode de "Branch and cut". Nous commençons par proposer de nouvelles contraintes valides pour le problème. Ensuite, nous les associons à des contraintes de couverture locales et globales dans un schéma énumératif. Les approches heuristiques et l'algorithme exact que nous proposons sont comparés à d'autres heuristiques de la littérature et au Solveur de programmes linéaires Cplex . L'ensemble de ces tests numériques ont été menés sur des instances ardues de la littérature ainsi que sur des instances générées aléatoirement de taille modérée.

La résolution jusqu'à l'optimalité de problèmes d'optimisation combinatoire NP-difficiles nécessite une mise en oeuvre de méthodes de plus en plus complexes qui consomment de plus en plus de puissance de calcul. L'objectif de notre travail est de paralléliser un algorithme de "Branch and Cut" pour résoudre jusqu'à l'optimalité des instances difficiles du voyageur de commerce. Dans la première partie de notre travail, nous présentons les composantes principales de l'algorithme du "Branch and Cut". Nous étudions ensuite le problème du voyageur de commerce par une approche polyédrale. Nous donnons enfin une description détaillée de notre implémentation de l'algorithme du "Branch and Cut". Dans la deuxième partie, Nous commençons par une brève présentation du parallélisme, et un état de l'art des études menées sur la parallélisation de l'algorithme du "Branch and Bound". Puis, nous proposons plusieurs modèles de parallélisations de l'algorithme du "Branch and Cut". Nous décrivons ensuite la stratégie de contrôle de la recherche arborescente qu'on a adopté, les mécanismes de minimisation des coûts liés aux différentes étapes de la communication entre les processeurs et les stratégies d'équilibrages. Nous terminons en donnant les résultats obtenus sur le IBM-SP1.

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.

The minimum k-partition (MkP) problem is a well-known optimization problem encountered in various applications most notably in telecommunication and physics. Formulated in the early 1990s by Chopra and Rao, the MkP problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in different partitions. In this thesis, we design and implement a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. We describe and study the properties of two relaxations of the MkP problem, the linear programming and the semidefinite programming relaxations. We then derive a new strengthened relaxation based on semidefinite programming. This new relaxation provides tighter bounds compared to the other two discussed relaxations but suffers in term of computational time. We further devise an iterative clustering heuristic (ICH), a novel heuristic that finds feasible solution to the MkP problem and we compare it to the hyperplane rounding techniques of Goemans and Williamson and Frieze and Jerrum for k=2 and for k=3 respectively. Our computational results support the conclusion that ICH provides a better feasible solution for the MkP. Furthermore, unlike the hyperplane rounding, ICH remains very effective in the presence of negative edge weights. Next we describe in detail the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) to find optimal solution for the MkP problem. The ICH heuristic is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-cut tree. Finally, we present computational results for the SBC algorithm on several classes of test instances with k=3, 5, and 7. Complete graphs with up to 60 vertices and sparse graphs with up to 100 vertices arising from a physics application were considered.

The minimum k-partition (MkP) problem is a well-known optimization problem encountered in various applications most notably in telecommunication and physics. Formulated in the early 1990s by Chopra and Rao, the MkP problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in different partitions. In this thesis, we design and implement a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. We describe and study the properties of two relaxations of the MkP problem, the linear programming and the semidefinite programming relaxations. We then derive a new strengthened relaxation based on semidefinite programming. This new relaxation provides tighter bounds compared to the other two discussed relaxations but suffers in term of computational time. We further devise an iterative clustering heuristic (ICH), a novel heuristic that finds feasible solution to the MkP problem and we compare it to the hyperplane rounding techniques of Goemans and Williamson and Frieze and Jerrum for k=2 and for k=3 respectively. Our computational results support the conclusion that ICH provides a better feasible solution for the MkP. Furthermore, unlike the hyperplane rounding, ICH remains very effective in the presence of negative edge weights. Next we describe in detail the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) to find optimal solution for the MkP problem. The ICH heuristic is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-cut tree. Finally, we present computational results for the SBC algorithm on several classes of test instances with k=3, 5, and 7. Complete graphs with up to 60 vertices and sparse graphs with up to 100 vertices arising from a physics application were considered.

This thesis deals with the exact solution of large-scale minimum bisection problems via a semidefinite relaxation in a branch-and-cut framework. After reviewing known results on the underlying bisection cut polytope a study of new facet-defining inequalities is presented. They are derived from the known knapsack tree inequalities. We investigate strengthenings based on the new cluster weight polytope and present polynomial separation algorithms for special cases. The dual of the semidefinite relaxation of the minimum bisection problem is tackled in its equivalent form as an eigenvalue optimisation problem with the spectral bundle method. Implementational details regarding primal heuristics, branching rules, so-called support extensions for cutting planes and warm start are presented. We conclude with a computational study in which we show that our approach is competetive to state-of-the-art implementations using linear programming or semidefinite programming relaxations. / Diese Dissertation befasst sich mit der exakten Lösung großer Minimum Bisection Probleme über eine semidefinite Relaxierung in einem Branch-and-Cut Zugang. Nachdem bekannte Resultate zum zugrundeliegenden Bisection Cut Polytop dargestellt wurden, wird eine Studie neuer facettendefinierender Ungleichungen präsentiert. Diese werden von den bekannten Knapsack Tree Ungleichungen abgeleitet. Wir untersuchen Verstärkungen basierend auf dem neuen Cluster Weight Polytop und zeigen polynomiale Separierungsalgorithmen für Spezialfälle. Die Duale der semidefiniten Relaxierung des Minumum Bisection Problems wird in ihrer äquivalenten Form als Eigenwertoptimierungsproblem mit dem Spektralen Bündelverfahren bearbeitet. Details der Implementierung bezüglich primaler Heuristiken, Branchingregeln, sogenannter Supporterweiterungen für die Schnittebenen und Warmstart werden präsentiert. Wir beenden die Arbeit mit einer numerischen Studie, in der wir zeigen, dass unser Zugang konkurrenzfähig zu aktuellen Implementationen basierend auf linearen und semidefiniten Relaxierungen ist.

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.

University, Diss., 1999--Saarbrücken.

Orientador: Vinicius Amaral Armentano / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-10T01:09:54Z (GMT). No. of bitstreams: 1 Fonseca_SaraLuisadeAndrade_M.pdf: 1363535 bytes, checksum: aa7c01c779a21ea25aa3b603425c92fe (MD5) Previous issue date: 2007 / Resumo: Em 1958, Gomory propôs uma desigualdade válida ou corte a partir do tableau do método simplex para programação linear, que foi utilizado no primeiro método genérico para resolução de problemas de programação inteira. Em 1960, o corte foi estendido para problemas de programação inteira mista. Em 1973, Chvátal sugeriu um corte derivado da formulação original do problema de programação inteira, e devido à equivalência com o corte de Gomory, este passou a ser chamado de corte de Chvátal-Gomory. A importância do corte de Gomory só foi reconhecida em 1996 dentro do contexto do método branch-and-cut para resolução de problemas de programação inteira e programação inteira mista. Desde então, este corte é utilizado em resolvedores comerciais de otimização. Recentemente, diversos cortes novos derivados do corte de Chvátal-Gomory foram propostos na literatura para programação inteira. Este trabalho trata do desenvolvimento de algoritmos para alguns destes cortes, e implementação computacional em um contexto de branch-and-cut, no ambiente do resolvedor CPLEX. A eficácia dos cortes é testada em instâncias dos problemas da mochila multidimensional, designação generalizada e da biblioteca MIPLIB. / Abstract: In 1958, Gomory proposed a valid inequality or cut from the tableau of the simplex method for linear programming, which was used in the first generic method for solving integer programming problems. In 1960, the cut was extended to handle mixed integer programming problems. In 1973, Chvátal suggested a cut that is generated from the original formulation of an integer programming problem, and due to the equivalence with the Gomory cut, it was named Chvátal-Gomory cut. The importance of the Gomory cut was recognized only in 1996 in the context of the branch-and-cut method for solving (mixed) integer programming problems. Today, such a cut is utilized in optimization commercial solvers. Recently, several new cuts derived from the Chvátal-Gomory cut have been proposed in the literature for integer programming. This work deals with the development of algorithms and computational implementations for some of the new proposed cuts, in a context of the branch-and-cut method, by using the CPLEX solver. The efficiency of the cuts is tested on instances of the multi-dimensional knapsack, generalized assignment problems, and instances from the MIPLIB library. / Mestrado / Automação / Mestre em Engenharia Elétrica