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

Estudi de la coautoria de publicacions científiques : Universidade Estadual de Campinas – UNICAMP Universitat Politècnica de Catalunya – UPC-BarcelonaTech

S'analitza la coautoria de la UPC amb autors vinculats a la Universidade Estadual de Campinas, per totes les àrees temàtiques, durant el període 2009-2013.Postprint (published version

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

On geometrical properties of preconditioners in IPMs for classes of block-angular problems

J. Castro, S. Nasini, On geometrical properties of preconditioners in IPMs for classes of block-angular problems, Research Report DR 2016/03, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, 2016.One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraints structure. In particular, it is seen that the principal angles between the subspaces generated by the diagonal blocks and the linking constraints can be used to estimate ex-ante the efficiency of the preconditioner. Numerical validation is provided with some generated optimization problems. An application to the solution of multicommodity network flow problems with nodal capacities and equal flows of up to 127 million of variables and up to 7.5 million of constraints is also presentedPreprin

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 Reviewe

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

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)The computational time required by interior-point methods is often dominated by the solution of linear systems of equations. An efficient specialized interior-point algorithm for primal block-angular problems has been used to solve these systems by combining Cholesky factorizations for the block constraints and a conjugate gradient based on a power series preconditioner for the linking constraints. In some problems this power series preconditioner resulted to be inefficient on the last interior-point iterations, when the systems became ill-conditioned. In this work this approach is combined with a splitting preconditioner based on LU factorization, which works well for the last interior-point iterations. Computational results are provided for three classes of problems: multicommodity flows (oriented and nonoriented), minimum-distance controlled tabular adjustment for statistical data protection, and the minimum congestion problem. The results show that, in most cases, the hybrid preconditioner improves the performance and robustness of the interior-point solver. In particular, for some block-angular problems the solution time is reduced by a factor of 10. (C) 2013 Elsevier B.V. All rights reserved.2312263273Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Spanish research program [MTM2009-08747, MTM2012-31440]Government of Catalonia [SGR-2009-1122]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Spanish research program [MTM2009-08747, MTM2012-31440]Government of Catalonia [SGR-2009-1122

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 Reviewe

Interior point methods iteration reduction with continued iteration

Orientadores: Aurelio Ribeiro Leite de Oliveira, Carla Taviane Lucke da Silva GhidiniTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Os métodos de pontos interiores têm sido extensivamente utilizado para resolver os problemas de programação linear de grande porte. Entre todas as variações de métodos de pontos interiores, o preditor corretor com múltiplas correções de centralidade apresenta um grande destaque, devido a sua eficiência e rápida convergência. Este método, necessita resolver sistemas lineares, em cada iteração, para determinar a direção de busca, correspondendo ao passo que requer mais tempo de processamento. Neste trabalho, a iteração continuada é apresentada e introduzida ao método preditor corretor com múltiplas correções de centralidade, com objetivo reduzir o número de iterações e o tempo computacional deste método para determinar a solução de problemas de programação linear. A iteração continuada consiste em determinar uma nova direção combinada com a direção de busca dos métodos de pontos interiores. Apresentamos duas novas direções continuadas e duas formas diferentes de utilizá-las, propondo um aumento no tamanho dos passos a serem dados na direção de busca, acelerando a convergência do método. Além disso, utilizamos o algoritmo de ajustamento ótimo para p coordenadas para determinar melhores pontos iniciais para o método de pontos interiores em conjunto com a iteração continuada. Experimentos computacionais foram realizados e os resultados obtidos ao incorporar a iteração continuada com o método de pontos interiores preditor corretor e as múltiplas correções de centralidade são superiores à abordagem tradicional. A utilização do algoritmo de ajustamento ótimo para p coordenadas na nova abordagem leva a resultados semelhantesAbstract: The interior point methods have been extensively used to solve large-scale linear programming problems. Among all variations of interior point methods, the predictor corrector with multiple centrality corrections is the method of choice due to its efficiency and fast convergence. This method requires solving linear systems, to determine the search direction corresponding to the step that requires more processing time, in each iteration. In this work, the continued iteration is presented and introduced to the predictor corrector method with multiple centrality corrections, in order to reduce the number of iterations and the computational time to determine the linear programming problems solution. The continued iteration consists of determining a new direction combined with the search direction of the interior point methods. Two new continued directions and two different ways of being used, increasing of the steps sizes taken in the search direction, speeding up the convergence of the method. In addition, we use the optimal adjustment algorithm for p coordinates to determine the best starting point for the interior point method in conjunction with the continued iteration. Computational experiments were performed and the results achieved by incorporating the continued iteration in the predictor corrector interior point method and multiple centrality corrections outperform the traditional approach. Using the optimal adjustment algorithm for p coordinates leads to similar resultsDoutoradoMatematica AplicadaDoutora em Matemática Aplicada2011/20623-7FAPES

Incomplete Cholesky factorizations for the direct solution of linear systems arising from interior point methods

Orientador: Aurelio Ribeiro Leite de OliveiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Uma das abordagens utilizadas para resolver o sistema linear que surge a cada iteração dos métodos de pontos interiores do tipo primal-dual é reduzi-lo a um sistema linear equivalente simétrico definido positivo, conhecido como sistema de equações normais, e aplicar a fatoração de Cholesky na matriz do sistema. A desvantagem desta abordagem é o preenchimento gerado durante a fatoração, o que pode tornar seu uso inviável por limitação de tempo e memória computacional. Neste trabalho, propomos um método que resolve de forma direta, sistemas lineares que se aproximam do sistema de equações normais e que exerce um certo controle sobre o preenchimento. Nossa proposta é na resolução direta deste sistema, substituir a fatoração de Cholesky por uma fatoração incompleta de Cholesky. A ideia é calcular, nas iterações iniciais, soluções aproximadas por meio de sistemas lineares cujas matrizes são fatores incompletos de Cholesky o mais esparsos possíveis. E nas iterações finais, calcular matrizes próximas ou iguais a fatoração de Cholesky completa, de forma que a convergência do método não seja afetada. Experimentos computacionais mostram que a abordagem proposta reduz de forma significativa o tempo de solução dos sistemas lineares nas iterações iniciais dos métodos de pontos interiores, levando a uma redução no tempo total de processamento de grande parte dos problemas testadosAbstract: One of the most commonly used approaches to solve the normal equation systems arising in primal-dual interior point methods is the direct solution by using the Cholesky factorization of the matrix system. The major disadvantage of this approach is the fill-in, which can make its use impracticable, due to time and memory limitations. In this work, we propose a method that directly solves an approximated system of normal equation keeping the fill-in under control. In our proposal, in the normal equation system direct solution, we replace the Cholesky factorization by an incomplete Cholesky factorization. The idea is to compute approximate solutions in the early iterations by linear systems whose matrices are incomplete Cholesky factors as sparses as possible and in the final iterations, to compute matrices close or equal to the complete Cholesky factorization so that the method convergence is kept. Computational experiments show that the proposed approach significantly reduces the linear systems solution time in the interior points methods in early iterations, leading to a reduction in the total processing time for many of the tested problemsDoutoradoMatematica AplicadaDoutora em Matemática Aplicada2013/02089-9FAPESPCAPE