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

Abstract

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.

Accelerated multigrid for graph Laplacian operators

“…A final important challenge would be the extension of our proposal to a non-symmetric setting, as it occurs when dealing with the celebrated Google problem [46,25,39], and to system matrices having structure of graph combined with structures of different nature, as it occurs when dealing with IP methods for problems related to more general graph-structured Linear or Quadratic Programs [7,15,16].…”

“…relation (2)). Not only the networks can be very large (e.g., with up to 2 22 arcs [54]), but these approaches allow a more or less direct extension [13,14] to multicommodity flow problems [30,15,16], that have a huge range of practical applications from telecommunication [18] to transportation [31] and beyond. In the latter setting, the size of the matrix is further multiplied by the number of commodities (different types of flows in the network), that can be easily run into the thousands (e.g.…”

Randomized rounding algorithms for large scale unsplittable flow problems

“…Another method presented in (Babonneau et al 2006) consists of a specialized interior-point method to solve the multi-commodity flow problem. This method has been improved by Castro and Cuesta (2012). Other contributions to linear programming methods can be found in Moradi et al (2015), Dai et al (2016a) and Dai et al (2016b) For large instances, linear programming methods may take a lot of computing time before finding the optimal solution.…”

Large-scale optimization with the primal-dual column generation method

“…These problems are usually solved by approaches which are based on column generation procedures [5,6,49]. As recognized in [5], there is a different type of MCNF problems in which the costs depend additionally on the commodities assigned to the arc, and the commodities compete for mutual and/or individual capacities [15,16,24]. No paper in the MCNF literature deals with both types of problems simultaneously.…”