We study a generic minimization problem with separable non-convex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.

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abstract: Carbon Capture and Storage (CCS) is a climate stabilization strategy that prevents CO2 emissions from entering the atmosphere. Despite its benefits, impactful CCS projects require large investments in infrastructure, which could deter governments from implementing this strategy. In this sense, the development of innovative tools to support large-scale cost-efficient CCS deployment decisions is critical for climate change mitigation. This thesis proposes an improved mathematical formulation for the scalable infrastructure model for CCS (SimCCS), whose main objective is to design a minimum-cost pipe network to capture, transport, and store a target amount of CO2. Model decisions include source, reservoir, and pipe selection, as well as CO2 amounts to capture, store, and transport. By studying the SimCCS optimal solution and the subjacent network topology, new valid inequalities (VI) are proposed to strengthen the existing mathematical formulation. These constraints seek to improve the quality of the linear relaxation solutions in the branch and bound algorithm used to solve SimCCS. Each VI is explained with its intuitive description, mathematical structure and examples of resulting improvements. Further, all VIs are validated by assessing the impact of their elimination from the new formulation. The validated new formulation solves the 72-nodes Alberta problem up to 7 times faster than the original model. The upgraded model reduces the computation time required to solve SimCCS in 72% of randomly generated test instances, solving SimCCS up to 200 times faster. These formulations can be tested and then applied to enhance variants of the SimCCS and general fixed-charge network flow problems. Finally, an experience from testing a Benders decomposition approach for SimCCS is discussed and future scope of probable efficient solution-methods is outlined. / Dissertation/Thesis / Masters Thesis Industrial Engineering 2017

Depuis une vingtaine d’années, la résolution de problèmes d’optimisation globale non convexes avec contraintes a connu un formidable essor. Les algorithmes de branch and bound basée sur l’analyse d’intervalles ont su trouver leur place, car ils ont l’avantage de prouver l’optimalité de la solution de façon déterministe, avec un niveau de certitude pouvant aller jusqu’à la précision machine. Cependant, la complexité exponentielle en temps et en mémoire de ces algorithmes induit une limite intrinsèque, c’est pourquoi il est toujours nécessaire d’améliorer les techniques actuelles. Dans cette thèse, nous avons développé de nouvelles arithmétiques basées sur l’arithmétique d’intervalles et l’arithmétique affine, afin de calculer des minorants et des majorants de meilleure qualité de fonctions explicites sur un intervalle. Nous avons ensuite développé une nouvelle méthode automatique de construction de relaxations linéaires. Cette construction est basée sur l’arithmétique affine et procède par surcharge des opérateurs. Les programmes linéaires ainsi générés ont exactement le même nombre de variables et de contraintes d’inégalité que les problèmes originaux, les contraintes d’égalité étant remplacées par deux inégalités. Cette nouvelle procédure permet de calculer des minorants fiables et des certificats d’infaisabilité pour chaque sous-domaine à chaque itération de notre algorithme de branch and bound par intervalles. De nombreux tests numériques issus du site COCONUT viennent confirmer l’efficacité de cette approche. Un autre aspect de cette thèse a été l’étude d’une extension de ce type d’algorithmes en introduisant une limite sur mémoire disponible. L’idée principale de cette approche est de proposer un processus inverse de l’optimisation par le biais d’un principe métaheuristique : plutôt que d’améliorer des solutions locales à l’aide de métaheuristiques telles que les algorithmes Taboo ou VNS, nous partons d’une méthode exacte et nous la modifions en une heuristique. De cette façon, la qualité de la solution trouvée peut être évaluée. Une étude de la complexité de ce principe métaheuristique a également été effectuée. Enfin, pour finir l’étude, nous avons appliqué notre algorithme à la résolution de problème en géométrie plane, ainsi qu’à la résolution d’un problème de dimensionnement de moteur électrique. Les résultats obtenus ont permis de confirmer l’intérêt de ce type d’algorithme, en résolvant des problèmes ouverts sur les polygones convexes et proposant des structures innovantes en génie électrique. / Since about thirty years, interval Branch and Bound algorithms are increasingly used to solve constrained global optimization problems in a deterministic way. Such algorithms are reliable, i.e., they provide an optimal solution and its value with guaranteed bounds on the error, or a proof that the problem under study is infeasible. Other approaches to global optimization, while useful and often less time-consuming than interval methods, do not provide such a guarantee. However, the exponential complexity in time and memory of interval Branch and Bound algorithms implies a limitation, so it is always necessary to improve these methods. In this thesis, we have developed new arithmetics based on interval arithmetic and affine arithmetic, to compute better lower and upper bounds of a factorable function over an interval. An automatic method for constructing linear relaxations of constrained global optimization problems is proposed. Such a construction is based on affine and interval arithmetics and uses operator overloading. These linear programs have exactly the same numbers of variables and of inequality constraints as the given problems. Each equality constraint is replaced by two inequalities. This new procedure for computing reliable bounds and certificates of infeasibility is inserted into a classical interval Branch and Bound algorithm. Extensive computation experiments, made on a sample of test problems from the COCONUT database, prove its effectiveness. Another aspect is the study of an extension of such a global optimization code by limiting the available memory. The main idea of this new kind of metaheuristique is to propose a reverse process of optimization via heuristics : rather than improving local solutions by using metaheuristics such as Taboo or VNS, we begin with an exact method and we modify it into a heuristic one. In such a way, the quality of the solution could be evaluated. Moreover, a study of the complexity of this metaheurisque has been done. Finally, we applied our algorithm to solve open problem in geometry, and to solve a design problem of an electric motor. The results have confirmed the usefulness of this kind of algorithms, solving open problems on convex polygons and offering innovative structures in electrical engineering.

Nesta tese propõe-se um método heurístico para o problema de Projeto de Rede da Cadeia de Suprimentos (Supply Chain Network Design) considerando vários aspectos de relevância prática, tais como: fornecedores e matérias-primas, localização e operação de instalações, atribuição de Centros de Distribuição (CD), e grande número de clientes e produtos. Uma eficiente abordagem heurística de duas fases é proposta para a obtenção de soluções viáveis para os problemas, que inicialmente é modelado como um Programa Linear Inteiro Misto (PLIM) de grande escala. Na fase de construção, uma estratégia de Linear Programming Rounding é aplicada para se obter os valores iniciais para as variáveis de localização inteira do modelo. Simultaneamente, um método Multi-start foi desenvolvido para gerar soluções iniciais diversificadas para cada nova iteração da heurística de Rounding. Na segunda fase, dois procedimentos de Busca Local foram desenvolvidos no sentido de melhorar a solução fornecida pelo método de Rounding. Implementamos duas diferentes abordagens de Busca Local: remoção-inserção e troca. Uma técnica de Busca Tabu para orientar o procedimento de Busca Local para explorar os diferentes espaços de soluções foi desenvolvida. As formulações e algoritmos foram implementados na linguagem C++ utilizando ferramentas de otimização da COIN-OR. O método de solução foi experimentado em instâncias geradas aleatoriamente, com tamanhos diferentes em termos do número de parâmetros, tais como o número de produtos, zonas de clientes, CDs e fábricas considerando um sistema logístico de quatro níveis. As implementações computacionais mostram que o método de solução proposto obteve resultados satisfatórios quando comparados com a literatura. Para validar este método heurístico também foi usado em um caso realista, com base em dados de uma empresa de borracha que está reestruturando sua cadeia de suprimentos devido ao projeto de uma nova uma nova fábrica e produção de novos produtos. A abordagem heurística proposta revelou-se adequada para aplicação prática em um caso real de uma indústria multicommodity em um contexto determinístico. / In this thesis we propose a heuristic method for the Supply Chain Network Design (SCND) problem considering several aspects of practical relevance: suppliers and raw materials, location and operation facilities, distribution center (DC) assignments, and large numbers of customers and products. An efficient two-phase heuristic approach is proposed for obtaining feasible solutions to the problems, which is initially modeled as a large-scale Mixed Integer Linear Program (MILP). In the construction phase, a linear programming rounding strategy is applied to obtain initial values for the integer location variables in the model. Simultaneously, a Multi-start method was developed to generate diversified initial solutions from each new iteration in the rounding heuristic. In the second phase, two Local Search procedures were developed towards to improve the solution provided by the rounding method. We implemented two different Local Search approaches: removal-insertion and exchange. A Tabu Search technique was developed to guide the Local Search procedure to explore the different spaces of solutions. The formulations and algorithms were implemented in C++ code language using the optimization engine COIN-OR. The solution method was experimented in randomly generated instances, with different sizes in terms of the number of parameters, such as number of products, customer zones, DCs, and factories considering a four-echelon logistic system. The computational implementations show that the solution method proposed obtained satisfactory results when compared to the literature review. To validate this heuristic method was also used in a realistic case, based on data from a rubber company that is restructuring its supply chain due to the overture of a new factory, producing new products. The proposed heuristic approach proved appropriate to practical application in a realistic case of a multi commodity industry in a deterministic context.

Nesta tese propõe-se um método heurístico para o problema de Projeto de Rede da Cadeia de Suprimentos (Supply Chain Network Design) considerando vários aspectos de relevância prática, tais como: fornecedores e matérias-primas, localização e operação de instalações, atribuição de Centros de Distribuição (CD), e grande número de clientes e produtos. Uma eficiente abordagem heurística de duas fases é proposta para a obtenção de soluções viáveis para os problemas, que inicialmente é modelado como um Programa Linear Inteiro Misto (PLIM) de grande escala. Na fase de construção, uma estratégia de Linear Programming Rounding é aplicada para se obter os valores iniciais para as variáveis de localização inteira do modelo. Simultaneamente, um método Multi-start foi desenvolvido para gerar soluções iniciais diversificadas para cada nova iteração da heurística de Rounding. Na segunda fase, dois procedimentos de Busca Local foram desenvolvidos no sentido de melhorar a solução fornecida pelo método de Rounding. Implementamos duas diferentes abordagens de Busca Local: remoção-inserção e troca. Uma técnica de Busca Tabu para orientar o procedimento de Busca Local para explorar os diferentes espaços de soluções foi desenvolvida. As formulações e algoritmos foram implementados na linguagem C++ utilizando ferramentas de otimização da COIN-OR. O método de solução foi experimentado em instâncias geradas aleatoriamente, com tamanhos diferentes em termos do número de parâmetros, tais como o número de produtos, zonas de clientes, CDs e fábricas considerando um sistema logístico de quatro níveis. As implementações computacionais mostram que o método de solução proposto obteve resultados satisfatórios quando comparados com a literatura. Para validar este método heurístico também foi usado em um caso realista, com base em dados de uma empresa de borracha que está reestruturando sua cadeia de suprimentos devido ao projeto de uma nova uma nova fábrica e produção de novos produtos. A abordagem heurística proposta revelou-se adequada para aplicação prática em um caso real de uma indústria multicommodity em um contexto determinístico. / In this thesis we propose a heuristic method for the Supply Chain Network Design (SCND) problem considering several aspects of practical relevance: suppliers and raw materials, location and operation facilities, distribution center (DC) assignments, and large numbers of customers and products. An efficient two-phase heuristic approach is proposed for obtaining feasible solutions to the problems, which is initially modeled as a large-scale Mixed Integer Linear Program (MILP). In the construction phase, a linear programming rounding strategy is applied to obtain initial values for the integer location variables in the model. Simultaneously, a Multi-start method was developed to generate diversified initial solutions from each new iteration in the rounding heuristic. In the second phase, two Local Search procedures were developed towards to improve the solution provided by the rounding method. We implemented two different Local Search approaches: removal-insertion and exchange. A Tabu Search technique was developed to guide the Local Search procedure to explore the different spaces of solutions. The formulations and algorithms were implemented in C++ code language using the optimization engine COIN-OR. The solution method was experimented in randomly generated instances, with different sizes in terms of the number of parameters, such as number of products, customer zones, DCs, and factories considering a four-echelon logistic system. The computational implementations show that the solution method proposed obtained satisfactory results when compared to the literature review. To validate this heuristic method was also used in a realistic case, based on data from a rubber company that is restructuring its supply chain due to the overture of a new factory, producing new products. The proposed heuristic approach proved appropriate to practical application in a realistic case of a multi commodity industry in a deterministic context.

Nesta tese propõe-se um método heurístico para o problema de Projeto de Rede da Cadeia de Suprimentos (Supply Chain Network Design) considerando vários aspectos de relevância prática, tais como: fornecedores e matérias-primas, localização e operação de instalações, atribuição de Centros de Distribuição (CD), e grande número de clientes e produtos. Uma eficiente abordagem heurística de duas fases é proposta para a obtenção de soluções viáveis para os problemas, que inicialmente é modelado como um Programa Linear Inteiro Misto (PLIM) de grande escala. Na fase de construção, uma estratégia de Linear Programming Rounding é aplicada para se obter os valores iniciais para as variáveis de localização inteira do modelo. Simultaneamente, um método Multi-start foi desenvolvido para gerar soluções iniciais diversificadas para cada nova iteração da heurística de Rounding. Na segunda fase, dois procedimentos de Busca Local foram desenvolvidos no sentido de melhorar a solução fornecida pelo método de Rounding. Implementamos duas diferentes abordagens de Busca Local: remoção-inserção e troca. Uma técnica de Busca Tabu para orientar o procedimento de Busca Local para explorar os diferentes espaços de soluções foi desenvolvida. As formulações e algoritmos foram implementados na linguagem C++ utilizando ferramentas de otimização da COIN-OR. O método de solução foi experimentado em instâncias geradas aleatoriamente, com tamanhos diferentes em termos do número de parâmetros, tais como o número de produtos, zonas de clientes, CDs e fábricas considerando um sistema logístico de quatro níveis. As implementações computacionais mostram que o método de solução proposto obteve resultados satisfatórios quando comparados com a literatura. Para validar este método heurístico também foi usado em um caso realista, com base em dados de uma empresa de borracha que está reestruturando sua cadeia de suprimentos devido ao projeto de uma nova uma nova fábrica e produção de novos produtos. A abordagem heurística proposta revelou-se adequada para aplicação prática em um caso real de uma indústria multicommodity em um contexto determinístico. / In this thesis we propose a heuristic method for the Supply Chain Network Design (SCND) problem considering several aspects of practical relevance: suppliers and raw materials, location and operation facilities, distribution center (DC) assignments, and large numbers of customers and products. An efficient two-phase heuristic approach is proposed for obtaining feasible solutions to the problems, which is initially modeled as a large-scale Mixed Integer Linear Program (MILP). In the construction phase, a linear programming rounding strategy is applied to obtain initial values for the integer location variables in the model. Simultaneously, a Multi-start method was developed to generate diversified initial solutions from each new iteration in the rounding heuristic. In the second phase, two Local Search procedures were developed towards to improve the solution provided by the rounding method. We implemented two different Local Search approaches: removal-insertion and exchange. A Tabu Search technique was developed to guide the Local Search procedure to explore the different spaces of solutions. The formulations and algorithms were implemented in C++ code language using the optimization engine COIN-OR. The solution method was experimented in randomly generated instances, with different sizes in terms of the number of parameters, such as number of products, customer zones, DCs, and factories considering a four-echelon logistic system. The computational implementations show that the solution method proposed obtained satisfactory results when compared to the literature review. To validate this heuristic method was also used in a realistic case, based on data from a rubber company that is restructuring its supply chain due to the overture of a new factory, producing new products. The proposed heuristic approach proved appropriate to practical application in a realistic case of a multi commodity industry in a deterministic context.

Les problèmes que nous considérons dans cette thèse sont de nature combinatoire. Notre principal intérêt est le problème de restructuration de données semi-structurées. Par exemple des données stockées sous la forme d’un fichier XML sont des données semi-structurées. Ce problème peut être ramené à une instance du problème du p-médian. Le principal obstacle ici est la taille des instances qui peut devenir très grande. Certaines instances peuvent avoir jusqu’à 10000 ou 20000 sommets, ce qui implique plusieurs centaines de millions de variables. Pour ces instances, résoudre ne serait-ce que la relaxation linéaire du problème est très difficile. Lors d’expériences préliminaires nous nous sommes rendu compte que CPLEX peut résoudre des instances avec 1000 sommets dans des temps raisonnables. Mais pour des instances de 5000 sommets, il peut prendre jusqu’à 14 jours pour résoudre uniquement la relaxation linéaire. Pour ces raisons nous ne pouvons utiliser de méthodes qui considère la résolution de la relaxation linéaire comme une opération de base, comme par exemple les méthodes de coupes et de branchements. Au lieu d’utiliser CPLEX nous utilisons une implémentation parallèle (utilisant 32 processeurs) de l’algorithme du Volume. L’instance pour laquelle CPLEX demande 14 heures est résolue en 24 minutes par l’implémentation séquentielle et en 10 minutes par l’implémentation parallèle de l’algorithme du Volume. La solution de la relaxation linéaire est utilisée pour construire une solution réalisable, grâce à l’application d’une heuristique de construction gloutonne puis d’une recherche locale. Nous obtenons des résultats comparables aux résultats obtenus par les meilleures heuristiques connues à ce jour, qui utilisent beaucoup plus de mémoire et réalisent beaucoup plus d’opérations. La mémoire est importante dans notre cas, puisque nous travaillons sur des données de très grandes tailles. Nous étudions le dominant du polytope associé au problème du p-médian. Nous discutons de sa relaxation linéaire ainsi que de sa caractérisation polyédrale. Enfin, nous considérons une version plus réaliste du problème de restructuration de données semi-structurées. Grosso modo, nous ajoutons au problème du p-médian original des nouveaux sommets s’ils aident à réduire le coût global des affectations. / The problems we consider in this thesis are of combinatorial nature. Our main interest is the problem of approximating typing of a semistructured data. For example XML is a semistructured data. This problem may be reduced to an instance of the p-median problem. The main obstacle here is the size of the instances that may be very huge, about 10000 and 20000 nodes which imply several hundreds of million variables. For these instances, even solving the linear relaxation is a hard task. In some preliminary results we noticed that Cplex may solve instances of size 1000 in an acceptable time. But for some instances having 5000 nodes, it may needs 14 days for solving only the linear relaxation. Therefore, we cannot use methods that consider the linear relaxation as an elementary operation, as for example branch-and-cut methods. Instead of using Cplex we use the Volume algorithm in a parallel implementation (32 processors).For the instance where the Cplex needs 14 hours, the Volume algorithm in sequential implementation needs 24 minutes and in parallel implementation it needs 10 minutes. The solution of the linear relaxation is used to produce a feasible solution by first applying a greedy and then a local search heuristic. We notice that the results we obtain are relatively the same as those given by the best method known up today, which produces more effort and consumes more memory. Memory is important in our case since the data we consider are huge. We study the dominant of the polytope associated with the p-median problem. We discuss linear relaxation and a polyhedral characterization. Finally, we consider a more realistic version of the p-median problem when applied to the problem of approximating typing of a semistructured data. Roughly speaking, we add new nodes to the underlying graph if this help to reduce the overall cost.