Made available in DSpace on 2014-07-29T16:02:20Z (GMT). No. of bitstreams: 1 Dissertacao Jose Carlos Rubianes Silva.pdf: 825326 bytes, checksum: f8797d1d8d333606ebad1d9941d5d26d (MD5) Previous issue date: 2011-02-18 / In this dissertation we consider a primal convex optimization problem and we study variants of subgradient method applied to the dual problem obtained via a Lagrangian function. We analyze the conditional subgradient method developed by Larsson et al, which is a variant of the usual subgradient method. In this variant, the subgradients are conditioned to a constraint set, more specifically, the behavior of the objective function outside of the constraint set is not taken into account. One motivation for studying such methods is primarily its simplicity, in particular, these methods are widely used in large-scale problems. The subgradient method, when applied to a dual problem, is relatively effective to obtain a good approximation of a dual solution and the optimal value, but it is not efficient to obtain primal solutions. We study a strategy to obtain good approximations of primal solutions via conditional subgradient method, under suitable additional computational costs. This strategy consists of constructing an ergodic sequence of solutions of the Lagrangian subproblems.We show that the limit points of this ergodic sequence are primal solutions. We consider different step sizes rule, in particular, following the ideas of Nedic and Ozdaglar, using the constant step size rule, we present estimates of the ergodic sequence and primal solutions and / or the feasible set. / Nesta dissertação consideramos um problema de otimização convexo e estudamos variações do método subgradiente aplicado ao problema dual obtido via uma função Lagrangiana. Estudamos o método subgradiente condicional desenvolvido por Larsson et al, o qual é uma simples variação do método subgradiente usual . A principal diferença é que os subgradientes são condicionados a um conjunto restrição, mais especificamente, o comportamento da função fora do conjunto restrição não é levado em conta. Uma motivação para estudar tais métodos consiste principalmente na sua simplicidade, em especial, estes métodos são bastante usados em problemas de grande porte. O método subgradiente, quando aplicado a um problema dual, é relativamente eficaz para obter boas aproximações de soluções duais e do valor ótimo, no entanto, não possue a mesma eficiência para obter soluções primais. Analisamos uma estratégia para obter boas aproximações de soluções primais via método subgradiente condicional, com pouco custo computacional adicional. Esta estratégia consiste em construir uma sequência ergódica das soluções obtidas durante a resolução dos subproblemas Lagrangianos. Mostraremos que os pontos limites desta sequência ergódica são soluções primais. Consideramos diferentes regras para o tamanho do passo, em particular, seguindo as idéias de Nedic e Ozdaglar, apresentamos estimativas da sequência ergódica com o conjunto de soluções primais e/ou o conjunto viável quando usamos a regra de passos constantes.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

The purpose of this research is to develop a model and solution methodology for selecting and scheduling timely and cost-effective maintenance, rehabilitation, and reconstruction activities (M & R) for each pavement section in a highway network and allocating the funding levels through a finite multi-period horizon within the constraints imposed by budget availability in each period, frequency availability of activities, and specified minimum pavement quality requirements. M & R is defined as a chronological sequence of reconstruction, rehabilitation, and major/minor maintenance, including a "do nothing" activity. A procedure is developed for selecting an M & R activity for each pavement section in each period of a specified extended planning horizon. Each activity in the sequence consumes a known amount of capital and generates a known amount of effectiveness measured in pavement quality. The effectiveness of an activity is the expected value of the overall gains in pavement quality rating due to the activity performed on a highway network over an analysis period. It is assumed that the unused portion of the budget for one period can be carried over to subsequent periods. Dynamic Programming (DP) and Branch-and-Bound (B-and-B) approaches are combined to produce a hybrid algorithm for solving the problem under consideratioin. The algorithm is essentially a DP approach in the sense that the problem is divided into smaller subproblems corresponding to each single period problem. However, the idea of fathoming partial solutions that could not lead to an optimal solution is incorporated within the algorithm to reduce storage and computational requirements in the DP frame using the B-and-B approach. The imbedded-state approach is used to reduce a multi-dimensional DP to a one-dimensional DP. For bounding at each stage, the problem is relaxed in a Lagrangean fashion so that it separates into longest-path network model subproblems. The values of the Lagrangean multipliers are found by a subgradient optimization method, while the Ford-Bellman network algorithm is employed at each iteration of the subgradient optimization procedure to solve the longest-path network problem as well as to obtain an improved lower and upper bound. If the gap between lower and upper bound is sufficiently small, then we may choose to accept the best known solutions as being sufficiently close to optimal and terminate the algorithm rather than continue to the final stage.

This study is concerned with the inventory routing problem with deterministic, dynamic demand and order-up-to level inventory policy. The problem mainly arises in the supply chain management context. It incorporates simultaneous decision making on inventory management and vehicle routing with the purpose of gaining advantage from coordinated decisions. An integrated mathematical model that represents the features of the problem is presented. Due to the magnitude of the model, lagrangean relaxation solution procedures that identify upper bounds and lower bounds for the problem are developed. Satisfactory computational results are obtained with the solution procedures suggested on the test instances taken from the literature.

In this thesis we study two discrete optimization problems: Traffic Light Synchronization and Location with Customers Orderings. A widely used approach to solve the synchronization of traffic lights on transport networks is the maximization of the time during which cars start at one end of a street and can go to the other without stopping for a red light (bandwidth maximization). The mixed integer linear model found in the literature, named MAXBAND, can be solved by optimization solvers only for small instances. In this manuscript we review in detail all the constraints of the original linear model, including those that describe all the cyclic routes in the graph, and we generalize some bounds for integer variables which so far had been presented only for problems that do not consider cycles. Furthermore, we summarized the first systematic algorithm to solve a simpler version of the problem on a single street. We also propose a solution algorithm that uses Tabu Search and Variable Neighbourhood Search and we carry out a computational study. In addition we propose a linear formulation for the shortest path problem with traffic lights constraints (SPTL). On the other hand, the simple plant location problem with order (SPLPO) is a variant of the simple plant location problem (SPLP) where the customers have preferences on the facilities which will serve them. In particular, customers define their preferences by ranking each of the potential facilities. Even though the SPLP has been widely studied in the literature, the SPLPO has been studied much less and the size of the instances that can be solved is very limited. In this manuscript, we propose a heuristic that uses a Lagrangean relaxation output as a starting point of a semi-Lagrangean relaxation algorithm to find good feasible solutions (often the optimal solution). We also carry out a computational study to illustrate the good performance of our method. Last, we introduce the partial and stochastic versions of SPLPO and apply the Lagrangean algorithm proposed for the deterministic case to then show examples and results.

DIAS, Fábio Carlos Sousa. Algoritmos para o problema de localização simples baseados nas formulações clássica e canônica. 2008. 81 f. : Dissertação (mestrado) - Universidade Federal do Ceará, Centro de Ciências, Departamento de Computação, Fortaleza-CE, 2008. / Submitted by guaracy araujo (guaraa3355@gmail.com) on 2016-06-22T17:13:52Z No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) / Approved for entry into archive by guaracy araujo (guaraa3355@gmail.com) on 2016-06-22T17:16:23Z (GMT) No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) / Made available in DSpace on 2016-06-22T17:16:23Z (GMT). No. of bitstreams: 1 2008_dis_fcsdias.pdf: 533140 bytes, checksum: 547c9cf8d771e2646884c423f5a39936 (MD5) Previous issue date: 2008 / In this work, we study the Simple Plant Location Problem (SPLP). Using its classical mathematical programming formulation and another recently proposed formulation, we develop several algorithms to …nd lower and upper bounds for the problem as well as branch-and-bound algorithms. With the classical formulation, such bounds are obtained via the data correction method and dominance criteria between …xed and transportation costs. We propose a projection of this formulation that has shown to be computationally atractive. Using the new formulation, we propose and prove the correctness of several iterative procedures that attempt to …nd an optimal solution to the problem by solving a sequence of parametric sub-problems, each one obtained by removing some variables and constraints of the original formulation. At each iteration of this process, we can obtain lower and upper bounds. We also apply Lagrangean relaxation to this new formulation in order to get other bounds. We consider several possibilities of relaxing the constraints. In addition, we develop branch-and-bound algorithms based on both formulations and the obtained bounds. We evaluate the computational e¢ ciency of all proposed algorithms with hard test instances from the literature. Computational results are reported and comparisons with other algorithms from the literature are carried out. / Neste trabalho, estudamos o problema de localização simples (SPLP - Simple Plant Location Problem). Usando a formulação matemática clássica e uma outra formulação proposta recentemente, desenvolvemos vários algoritmos para encontrar limites inferiores e superiores, bem como algoritmos tipo branch-and-bound. Com a formulação clássica, tais limites são obtidos utilizando o método de correção de dados e critérios de dominância entre os custos …xos e de transporte. Propomos uma projeção dessa formulação, que se mostrou computacionalmente atrativa. Usando a nova formulação propomos e mostramos a corretude de vários procedimentos iterativos que procuram encontrar uma solução para o problema, resolvendo uma seqüência de subproblemas paramétricos obtidos com a remoção de variáveis e restrições da formulação original. Em cada iteração desse processo, podemos gerar limites inferiores e superiores. Aplicamos ainda relaxação lagrangeana a essa nova formulação para obter outros limites. Analisamos várias possibilidades de relaxação das restrições. Desenvolmento também algoritmos branch-and-bound baseados em ambas as formulações e nos limites obtidos. Avaliamos a e…ciência computacional de todos os algoritmos com instâncias de teste difíceis, disponíveis na literatura. Resultados computacionais e comparações com outros algoritmos da literatura são reportados.

Orientadores: Secundino Soares Filho, Takaaki Ohishi / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-06T02:53:29Z (GMT). No. of bitstreams: 1 ArceEncina_AnastacioSebastian_D.pdf: 1529112 bytes, checksum: a6a22a9ce6cd536449dd06144205c2e5 (MD5) Previous issue date: 2006 / Resumo: A Programação diária de um sistema de energia elétrica busca obter um despacho de geração para o próximo dia, que seja compatível com as metas definidas pelo Planejamento energético e que sirva como referência operativa para a operação em tempo real. Assim, a modelagem do sistema deve ser detalhada levando em consideração as características dos sistemas de geração, de transmissão, requisitos do mercado, critérios de segurança e custos associados à operação. Este trabalho apresenta um modelo de despacho ótimo de unidades geradoras hidrelétricas que adota como critério de desempenho um modelo que avalia as perdas no sistema de geração, ocasionadas pela elevação do nível de canal de fuga, pela variação do rendimento do conjunto turbina-gerador e pelo atrito do fluxo d¿água nas tubulações do sistema hidráulico. Além das perdas no sistema de geração também forma parte do critério de desempenho o custo associado à partida e parada das unidades geradoras. Na formulação do problema, verifica-se a presença de variáveis inteiras, não lineares; restrições de atendimento à demanda, de meta de geração oriunda do Planejamento energético e restrições de capacidade de geração. Isto caracteriza o problema do despacho de unidades geradoras como sendo um problema de programação misto inteiro-não linear, combinatório e de grande porte quando considerado um sistema de geração como o do Sistema Elétrico Brasileiro. Como técnica de solução foi adotada uma heurística que combina Relaxação Lagrangeana e Programação Dinâmica. A metodologia foi aplicada a uma parte do sistema de geração do Sistema Elétrico Brasileiro, composta por 78 usinas hidrelétricas e o total de 398 unidades geradoras. Tomando como referência os dados de programação de um dia típico de operação, os resultados apontam ganhos significativos pela diminuição de perdas. Verifica-se também a importância de se considerar o custo das partidas e paradas no despacho das unidades geradoras / Abstract: The daily scheduling operation of the electrical energy system has by target to find the dispatch generation to the next day which must be compatible with the daily generation target defined by the energy planning models and to be an operational guide. Because of its proximity with the real time operation the modeling must represent the main characteristic of the generation and transmission systems, requirements market¿s, security criterion and operational costs. This work is concerned with the dispatch of hydro generating units. The performance criterion adopted takes into account variations in tailrace elevation, penstock head losses, turbine-generator efficiencies and start-up and shutdown costs. In the hydro generating units dispatch problem formulation there are integer, non-linear variables, constraints concerned with the generation capacity, load attainment and daily generation target, this last defined by the energy planning models. According above mentioned the dispatch of hydro generating units is a combinatorial and mixed integer non-linear programming problem whose solution can be quite complex for large hydroelectric system such as the Brazilian¿s generation system. A heuristic procedure based on Lagrangian Relaxation and Dynamic Programming is adopted as solution methodology. The approach has been tested on a hydro system composed of seventy and eight hydro plants with three hundred and ninety eight generating units of the Brazilian power generation system. The generation scheduling verified on a typical day was considered for comparison with the solution provided by the approach and the results show significant improvement in term of minimization looses and the importance of to consider the start-up and shut-down costs of hydro generating units / Doutorado / Energia Eletrica / Doutor em Engenharia Elétrica

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, estudamos o problema de localizaÃÃo simples (SPLP - Simple Plant Location Problem). Usando a formulaÃÃo matemÃtica clÃssica e uma outra formulaÃÃo proposta recentemente, desenvolvemos vÃrios algoritmos para encontrar limites inferiores e superiores, bem como algoritmos tipo branch-and-bound. Com a formulaÃÃo clÃssica, tais limites sÃo obtidos utilizando o mÃtodo de correÃÃo de dados e critÃrios de dominÃncia entre os custos …xos e de transporte. Propomos uma projeÃÃo dessa formulaÃÃo, que se mostrou computacionalmente atrativa. Usando a nova formulaÃÃo propomos e mostramos a corretude de vÃrios procedimentos iterativos que procuram encontrar uma soluÃÃo para o problema, resolvendo uma seqÃÃncia de subproblemas paramÃtricos obtidos com a remoÃÃo de variÃveis e restriÃÃes da formulaÃÃo original. Em cada iteraÃÃo desse processo, podemos gerar limites inferiores e superiores. Aplicamos ainda relaxaÃÃo lagrangeana a essa nova formulaÃÃo para obter outros limites. Analisamos vÃrias possibilidades de relaxaÃÃo das restriÃÃes. Desenvolmento tambÃm algoritmos branch-and-bound baseados em ambas as formulaÃÃes e nos limites obtidos. Avaliamos a e…ciÃncia computacional de todos os algoritmos com instÃncias de teste difÃceis, disponÃveis na literatura. Resultados computacionais e comparaÃÃes com outros algoritmos da literatura sÃo reportados. / In this work, we study the Simple Plant Location Problem (SPLP). Using its classical mathematical programming formulation and another recently proposed formulation, we develop several algorithms to …nd lower and upper bounds for the problem as well as branch-and-bound algorithms. With the classical formulation, such bounds are obtained via the data correction method and dominance criteria between …xed and transportation costs. We propose a projection of this formulation that has shown to be computationally atractive. Using the new formulation, we propose and prove the correctness of several iterative procedures that attempt to …nd an optimal solution to the problem by solving a sequence of parametric sub-problems, each one obtained by removing some variables and constraints of the original formulation. At each iteration of this process, we can obtain lower and upper bounds. We also apply Lagrangean relaxation to this new formulation in order to get other bounds. We consider several possibilities of relaxing the constraints. In addition, we develop branch-and-bound algorithms based on both formulations and the obtained bounds. We evaluate the computational e ciency of all proposed algorithms with hard test instances from the literature. Computational results are reported and comparisons with other algorithms from the literature are carried out.

[pt] Nesta tese são propostos modelos e algoritmos aproximados para o Problema da Árvore Geradora de Custo Mínimo com Restrição de Diâmetro (AGMD). Este problema modela tipicamente aplicações em projetos de redes de computadores onde todos os vértices devem comunicar-se entre si a um custo mínimo, garantindo um certo nível de serviço. Os modelos propostos por Achuthan e Caccetta para o AGMD são reforçados através da introdução de restrições válidas. Uma relaxação lagrangeana é proposta para o modelo de multifluxo básico de Gouveia e Magnanti. Essa relaxação é utilizada para o desenvolvimento de heurísticas lagrangeanas. Adaptações são realizadas nas heurísticas construtivas propostas por Deo e Abdalla, e por Raidl e Julstrom. São propostas ainda quatro estratégias de busca local, uma heurística do tipo GRASP e outra híbrida. São obtidos limites superiores a menos de 2% do ótimo para as classes de instâncias usadas nos trabalhos de Gouveia e Magnanti, e de Santos, Lucena e Ribeiro. Além disto, obteve-se os melhores resultados conhecidos até o presente momento para 11 instâncias de grafos completos usadas por Raidl, Julstrom e Gruber. / [en] In this work, models and approximation algorithms to solve the Diameter Constrained Minimum Spanning Tree Problem (AGMD) are proposed. This problem typically models network design applications where all vertices must communicate with each other at a minimum cost, while meeting a given quality requirement. The formulations proposed by Achuthan and Caccetta are strengthened with valid inequalities. A lagrangean relaxation is proposed for the multicommodity flow model developed by Gouveia and Magnanti. Adaptations are made in the constructive heuristics proposed by Deo and Abdalla and by Raidl and Julstrom. Four local search procedures, a GRASP algorithm and a hybrid heuristic are proposed. Upper bounds within 2% of the optimal solution values are obtained for the two classes of instances used by Gouveia and Magnanti and by Santos, Lucena and Ribeiro. Moreover, stronger upper bounds are reported for 11 instances in complete graphs used by Raidl, Julstrom and Gruber

Os derramamentos de petróleo são consequência inevitável e indesejável da produção e transporte do petróleo e seus derivados. A maioria desses derramamentos são relativamente pequenos, mas alguns deles são grandes o suficiente para causar significativo impacto ambiental. Nessas situações, os modelos computacionais são ferramentas importantes para estimar a trajetória, dimensionamento e comportamento do óleo derramado no ambiente marinho, sendo determinantes na elaboração de planos de ação e trabalho das equipes de resposta. O transporte e destino de óleo offshore derramado são regidos majoritariamente, no curto período, por processos de transporte e de transformação físico-químicos e no longo período por processos de degradação biológica, de acordo com as condições ambientais locais (oceânicas e atmosféricas). Os principais processos que atuam sobre as manchas de óleo offshore incluem, no curto período, advecção, difusão turbulenta, espalhamento superficial, evaporação, dissolução, emulsificação, sedimentação e a interação de mancha de óleo com a linha da costa. O STFM (Spill, Transport and Fate Model) foi o modelo computacional desenvolvido nesse trabalho. Os algoritmos foram desenvolvidos com base em formulações físico-químicas propostas na literatura, sendo testadas as proposições de diversos autores e selecionadas as equações que apresentaram melhores resultado para integrar o conjunto físico-químico que compõe o STFM. Os resultados do trabalho mostraram que o STFM apresentou desempenho superior aos demais modelos testados na descrição do espalhamento e difusão dando mais estabilidade à mancha por utilizar a derivação de Dodge para a proposta de espalhamento de Fay e substituir o método usual de Randon Walk por Randon Flight (avançado no tempo) na forma canônica dada por Lynch. O algoritmo do STFM também traz outra evolução importante ao incluir um modelo de evaporação baseado nas equações empíricas de Fingas, substituindo as atuais parametrizações baseadas no ADIOS2 e nos métodos de pseudocomponentes. / Oil and its by-products spills are an inevitable and undesirable consequence of their production and transportation. Even though these spills are relatively small, some of them are large enough to cause significant environmental impact. Taken this into account, the computational models are important tools to estimate the trajectory, dimensioning and behavior of the oil spilled in the marine environment, being also determinants to elaborate action plans for response teams work. The transportation and fate of oil spills are governed in the short term by physical-chemical transport and transformation processes and in the long term by biological degradation processes, according to local environmental conditions (oceanic and atmospheric). The main processes that act on offshore oil spills include, in the short term, advection, turbulent diffusion, surface scattering, evaporation, dissolution, emulsification, sedimentation and the interaction of oil slick according to the coast line. The Spill, Transport and Fate Model (STFM) was the computational model developed in this work. The algorithms were developed based on physicochemical formulations proposed in literature, being the propositions of several authors tested and the equations which presented the best results were selected to integrate the physical-chemical set that makes up the STFM. The STFM results presented superior performance, giving more stability to the stain, compared to the other models tested in the scattering and diffusion description, by using the Dodge derivation for the Fay spreading proposal and by replacing the usual \"Randon Walk\" method by \"Randon Flight\" (advanced in time) in the canonical form given by Lynch. The STFM algorithm also brings forward another important evolution by including an evaporation model based on Fingas empirical equations, replacing the current parameterizations based on ADIOS2 and pseudo component methods.