Improving an interior-point approach for large block-angular problems by hybrid preconditioners

The computational time required by interior-point methods is often domi- nated by the solution of linear systems of equations. An efficient spec ialized interior-point algorithm for primal block-angular proble ms has been used to solve these systems by combining Cholesky factorizations for the block con- straints and a conjugate gradient based on a power series precon ditioner for the linking constraints. In some problems this power series prec onditioner re- sulted to be inefficient on the last interior-point iterations, wh en the systems became ill-conditioned. In this work this approach is combi ned with a split- ting preconditioner based on LU factorization, which is main ly appropriate for the last interior-point iterations. Computational result s are provided for three classes of problems: multicommodity flows (oriented and no noriented), minimum-distance controlled tabular adjustment for statistic al data protec- tion, and the minimum congestion problem. The results show that , in most cases, the hybrid preconditioner improves the performance an d robustness of the interior-point solver. In particular, for some block-ang ular problems the solution time is reduced by a factor of 10.Peer ReviewedPreprin

Interior-point solver for convex separable block-angular problems

Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the precondi- tioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using l1 and l2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to l1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.Preprin

Quadratic regularizations in an interior-point method for primal block-angular problems

One of the most efficient interior-point methods for some classes of primal block-angular problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. Its efficiency depends on the spectral radius—in [0,1)— of a certain matrix in the definition of the preconditioner. Spectral radius close to 1 degrade the performance of the approach. The purpose of this work is twofold. First, to show that a separable quadratic regularization term in the objective reduces the spectral radius, significantly improving the overall performance in some classes of instances. Second, to consider a regularization term which decreases with the barrier function, thus with no need for an extra parameter. Computational experience with some primal block-angular problems confirms the efficiency of the regularized approach. In particular, for some difficult problems, the solution time is reduced by a factor of two to ten by the regularization term, outperforming state-of-the-art commercial solvers.Peer ReviewedPostprint (author’s final draft

Network Flows Heuristics for Complementary Cell Suppression: An Empirical Evaluation and Extensions

Several network flows heuristics have been suggested in the past for the solution of the complementary suppression problem. However, a limited computational experience using them is reported in the literature, and, moreover, they were only appropriate for two-dimensional tables. The purpose of this paper is twofold. First, we perform an em-pirical comparison of two network flows heuristics. They are improved versions of already existing approaches. Second, we show that exten-sions of network flows methods (i.e., multicommodity network flows and network flows with side constraints) can model three-dimensional, hierarchical and linked tables. Exploiting this network structure can improve the performance of any solution method solely based on linear programming formulations

Improving an interior-point algorithm for multicommodity flows by quadratic regularizations

One of the best approaches for some classes of multicommodity flow problems is a specialized interior-point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work the authors improved this algorithm (i.e., reduced the spectral radius) for general block-angular problems by adding a quadratic regularization to the logarithmic barrier. This barrier was shown to be self-concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work we focus on linear multicommodity problems, a particular case of primal block-angular ones. General results are tailored for multicommodity flows, allowing a local sensitivity analysis on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior-point algorithm is more efficient than the nonregularized one. From this work it can be concluded that, if interior-point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones.Preprin

Video-on-demand optimization using an interior-point algorithm

A Content Delivery Network (CDN) aims to provide efficient movement of massive digital content (multimedia and files) across the Internet. This is achieved by putting the content in servers closer to the costumer. Video-On-Demand service is an application of CDN where videos have to be located strategically to avoid network congestion and servers saturation. Therefore, the problem of optimal placement of videos arises. This problem has a block diagonal structure with linking constraints on links and servers capacities. In this project, we solve huge instances of a video placement problem over three real network topologies with a specialized interior point solver named BlockIP. The evaluated instances range from 7 to 300 millions of variables and the difficulty of the instances depends on the size of servers, links bandwidth and network topology. Our results: 1) verified characteristics of BlockIP like regularization and the intensive computation in the last iterations and 2) showed that BlockIP found optimal solution in all the evaluated instances with a good optimality gap. On the contrary, state-of-art CPLEX cannot reach an optimal, feasible solution in some difficult instances and needs almost twice the memory of BlockIP. However, CPLEX solved most of feasible instances at least twice faster than BlockI

Branch-and-price and multicommodity flows

Tese de doutoramento em Engenharia de Produção e Sistemas, área de Investigação OperacionalIn this Thesis, we address column generation based methods for linear and integer programming and apply them to three multicommodity flow problems. For (mixed) integer programming problems, the approach taken consists in reformulating an original model, using the Dantzig-Wolfe decomposition principle, and then combining column generation with branch-and-bound (branch-and-price) in order to obtain optimal solutions. The main issue when developing a branch-and-price algorithm is the branching scheme. The approach explored in this work is to branch on the variables of the original model, keeping the structure of the subproblems of the column generation method unchanged. The incorporation of cuts (branch-and-price-and-cut), again without changing the structure of the subproblem, is also explored. Based on that general methodology, we developed a set of C++ classes (ADDing - Automatic Dantzig-Wolfe Decomposition for INteger column Generation), which implements abranch-and-price algorithm. Its main distinctive feature is that it can be used as a “black-box”: all the user is required to do is to provide the original model. ADDing can also be customised to meet a specific problem, if the user is willing to provide a subproblem solver and/or specific branching schemes. We developed column generation based algorithms for three multicommodity flow problems. In this type of problems, it is desired to route a set of commodities through a capacitated network at a minimum cost. In the linear problem, each unit of each commodity is divisible. By using a model with variables associated with paths and circuits, we obtained significant improvements on the solution times over the standard column generation approach, for instances defined in planar networks (in several instances the relative improvement was greater than 60%). In the integer problem, each unit of each commodity is indivisible; the flow of a commodity can be split between different paths, but the flow on each of those paths must be integer. In general, the proposed branch-and-price algorithm was more efficient than Cplex 6.6 in the sets of instances where each commodity is defined by an origin-destination pair; for some of the other sets of instances, Cplex 6.6 gave better time results. In the binary problem, all the flow of each commodity must be routed along a single path. We developed a branch-and-price algorithm based on a knapsack decomposition and modified (by using a different branching scheme) a previously described branch-and-price-and-cut algorithm based on a path decomposition. The outcome of the computational tests was surprising, given that it is usually assumed that specific methods are more efficient than general ones. For the instances tested, a state-of-the-art general-purpose (Cplex 8.1) gave, in general, much better results than both decomposition approaches.Nesta Tese, abordam-se métodos de geração de colunas para programação linear e inteira. A sua aplicação é feita em três problemas de fluxo multicomodidade. Para problemas de programação inteira (mista), a abordagem seguida é a de reformular um modelo original, utilizando o princípio de decomposição de Dantzig-Wolfe, e combinar geração de colunas com o método de partição e avaliação (partição e geração de colunas) para a obtenção de soluções óptimas. A questão essencial no desenvolvimento de um algoritmo deste tipo é a estratégia de partição. A abordagem seguida neste trabalho é a de realizar a partição nas variáveis do modelo original, mantendo a estrutura do subproblema do método de geração de colunas. A incorporação de cortes, ainda sem alteração da estrutura do subproblema, é também explorada. Com base nesta metodologia geral, foi desenvolvido um conjunto de classes em C++ (ADDing - Automatic Dantzig-Wolfe Decomposition for INteger column Generation), que implementa um algorithmo de partição e geração de colunas. A sua característica fundamental é apenas ser requerido ao utilizador a definição de um modelo original. Num modo mais avançado, o utilizador pode implementar algoritmos para resolver o subproblema e/ou esquemas de partição. Foram desenvolvidos algoritmos baseados em geração de colunas para três problemas de fluxo multicomodidade. Neste tipo de problemas, pretende-se encaminhar um conjunto de comodidades através de uma rede capacitada, minimizando o custo. No problema linear, cada unidade de cada comodidade é divisível. Utilizando um modelo com variáveis associadas a caminhos e a circuitos, obtiveram-se melhorias significativas nos tempos de resolução em relação ao método de geração de colunas usual, para instâncias definidas em redes planares (em várias instâncias a melhoria relativa foi superior a 60%). No problema inteiro, cada unidade de cada comodidade é indivisível; o fluxo de uma comodidade pode ser dividido por diferentes caminhos, mas o fluxo em cada um deles tem de ser inteiro. Em geral, o algoritmo de partição e geração de colunas foi mais eficiente do que o software Cplex 6.6 nos conjuntos de instâncias em que cada comodidade é definida por um par origem-destino; para alguns dos outros conjuntos de instâncias, o software Cplex 6.6 obteve melhores resultados. No problema binário, todo o fluxo de cada comodidade apenas pode utilizar um caminho. Foi desenvolvido um algoritmo de partição e geração de colunas baseado numa decomposição de mochila e modificado (através de um esquema de partição diferente) um algoritmo de partição e geração de colunas com cortes, previamente descrito, baseado numa decomposição por caminhos. Os resultados dos testes computacionais foram surpreendentes, dado que é usualmente assumido que métodos específicos são mais eficientes do que métodos gerais. Para as instâncias testadas, o software Cplex 8.1 obteve, em geral, resultados muito melhores do que as duas decomposições

Um modelo de fluxo em rede para solução de problemas de distribuição de produtos compostos

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia de Produção.Neste trabalho é proposto um modelo linear de Fluxo em Redes para o problema de minimização de custos de produção e distribuição de Múltiplos Produtos Compostos. Neste modelo, restrições de acoplamento são consideradas para tratar a proporcionalidade existente entre os diversos fluxos que formam o produto composto, bem como as restrições de capacidade dos arcos pelos quais estes fluxos percorrem. A metodologia utilizada para solucionar o problema é baseada na estratégia de particionamento da matriz básica, e na implementação de uma especialização do método simplex dual para solucionar o problema particionado primal. Como solução inicial, é utilizada uma base construída por meio de um método heurístico que aloca fluxos em caminhos de custo mínimo. Para realização das operações de troca de base, a matriz ciclo é armazenada na forma produto da inversa, de modo a manter a esparsidade e a dimensão. Testes computacionais, contendo em torno de 200.000 restrições e 370.000 variáveis, aplicados à distribuição de produtos compostos de uma indústria do setor petroquímico, foram realizados com sucesso. Os resultados obtidos demonstram a eficiência computacional do algoritmo desenvolvido e a aplicabilidade do modelo formulado. Finalmente, recomendações são apresentadas para desenvolvimento de trabalhos futuros

Mathematical programming based approaches for classes of complex network problems : economical and sociological applications

The thesis deals with the theoretical and practical study of mathematical programming methodologies to the analysis complex networks and their application in economic and social problems. More specifically, it applies models and methods for solving linear and integer programming problems to network models exploiting the matrix structure of such models, resulting in efficient computational procedures and small processing time. As a consequence, it allows the study of larger and more complex networks models that arise in many economical and sociological applications. The main efforts have been addressed to the development of a rigorous mathematical programming based framework, which is able to capture many classes of complex network problems. Such a framework involves a general and flexible modeling approach, based on linear and integer programmin, as well as a collection of efficient probabilistic procedures to deal with these models. The computer implementation has been carried out by high level programming languages, such as Java, MatLab, R and AMPL. The final chapter of the thesis introduced an extension of the analyzed model to the case of microeconomic interaction, providing a fruitful mathematical linkage between its optimization-like properties and its multi-agents properties. The theoretical and practical use of optimization methods represents the trait-de-union of the different chapters. The overall structure of the thesis manuscript contains three parts: Part I: The fine-grained structure of complex networks: theories, models and methods; Chapter 1 and Chapter 2. Part II: Mathematical Programming based approaches for random models of network formation; Chapter 3, Chapter 4 and Chapter 5. Part III: Strategic models of network formation. Chapter 6. Results of this research have generated four working papers in quality scientific journals: one has been accepted and three are under review. Some results have been also presented in four international conferences.La tesis aborda el estudio teórico y práctico de las metodologías de programación matemática para el análisis de redes complejas y su aplicación a problemas económicos y sociales. Más específicamente, se aplica modelos y métodos para resolver problemas de programación lineal y de programación lineal entera explotando las estructuras matriciales de tales modelos, lo que resulta en procedimientos computacionales eficientes y bajo coste de procesamiento. Como consecuencia de ello, las metodologías propuestas permiten el estudio de modelos complejos de gran dimensión, para redes complejas que surgen en muchas aplicaciones económicas y sociológicas. Los principales esfuerzos se han dirigido al desarrollo de un marco teórico basado en la programación matemática, que es capaz de capturar muchas clases de problemas de redes complejas. Dicho marco teórico envuelve un sistema general y flexible de modelado y una colección de procedimientos probabilísticos para solucionar eficientemente dichos modelos, basados en la programación linear y entera. Las implementaciones informáticas se han llevado a cabo mediante lenguajes de programación de alto nivel, como Java, Matlab, R y AMPL. El último capítulo de la tesis introduce una extensión de los modelos analizados, para el caso de la interacción microeconómica, con el objetivo de establecer un nexo metodológico entre sus propiedades de optimización y sus propiedades multi-agentes. El uso teórico y práctico de los métodos de optimización representa el elemento de conjunción de los distintos capítulos. Parte I: The fine-grained structure of complex networks: theories, models and methods; - Capitulo 1 y Capitulo 2. Parte II: Mathematical Programming based approaches for random models of network formation; - Capitulo 3, Capitulo 4 y Capitulo 5. Parte III: Strategic models of network formation. - Capitulo 6. Los resultados de esta investigación han generado cuatro papers en revistas científicas indexadas: uno ha sido aceptado, tres están en revisión. Algunos resultados han sido también presentados en cuatro conferencias internacionale

Otimização do processo produtivo de um frigorífico de aves

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-graduação em Engenharia de ProduçãoOs frigoríficos de aves possuem algumas particularidades em relação aos processos produtivos convencionais de administração da produção, sendo a principal peculiaridade o fato de uma única matéria-prima ser "desmontada" ao longo da fábrica originando diversos produtos acabados. Além disso, esses produtos possuem características de commodity e estão inseridos em um mercado extremamente competitivo. Dentro dessas condições, a determinação do mix de produtos que deve ser fabricado diariamente pode ser considerada uma atividade estratégica dentro da empresa. Sendo assim, buscou-se neste trabalho o desenvolvimento de um modelo que visa otimizar o processo produtivo de um frigorífico de aves através da maximização da contribuição ao lucro dos produtos fabricados, utilizando ferramentas de programação matemática. Buscou-se, também, através da otimização do processo produtivo, a redução dos estoques de produtos acabados. Por fim, realizou-se um estudo de caso em uma empresa do setor para verificar a aplicabilidade do modelo proposto. Através dos resultados obtidos, algumas conclusões e possibilidades de desenvolvimento são apresentadas. The poultry production process in cold storage plants is different from conventional production process in that one, main raw material is "decomposed" throughout the plant to generate several different finished products. Furthermore, these products have commodity characteristics in very competitive markets. Because of this, the daily determination of the product mix to be produced should be considered a strategic activity of the company. This dissertation develops a model to optimize the production process of a poultry cold storage plant through the maximization of product's profit contribute, and use mathematical programming to solve the model. An expected by-product of the optimization of the production process was a reduction in finished goods inventory. The model was tested and validated through a case study and numerical example using a company in the poultry sector. Results and conclusions are presented and opportunities for further research are suggested