A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The effect of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. This approach allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour, outperforming other stateof- the-art methods, which exhausted the (144 Gigabytes of) available memory in the largest instances.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

Mathematical programming approaches for classes of random network problems

Random simulations from complicated combinatorial sets are often needed in many classes of stochastic problems. This is particularly true in the analysis of complex networks, where researchers are usually interested in assessing whether an observed network feature is expected to be found within families of networks under some hypothesis (named conditional random networks, i.e., networks satisfying some linear constraints). This work presents procedures to generate networks with specified structural properties which rely on the Solution of classes of integer optimization problems. We show that, for many of them, the constraints matrices are totally unimodular, allowing the efficient generation of conditional random networks by specialized interior-point methods. The computational results suggest that the proposed methods can represent a general framework for the efficient generation of random networks even beyond the models analyzed in this paper. This work also opens the posSibility for other applications of mathematical programming in the analysis of complex networks. (C) 2015 Elsevier B.V. All rights reserved.Peer Reviewe