Solving Splitted MultiCommodity Flow Problem by Efficient Linear Programming Algorithm
Abstract
Column generation is often used to solve multicommodity flow problems. A program for column generation always includes a module that solves a linear equation. In this paper, we address three major issues in solving linear problem during column generation procedure which are (1) how to employ the sparse property of the coefficient matrix; (2) how to reduce the size of the coefficient matrix; and (3) how to reuse the solution to a similar equation. To this end, we first analyze the sparse property of coefficient matrix of linear equations and find that the matrices occurring in iteration are very sparse. Then, we present an algorithm locSolver (for localized system solver) for linear equations with sparse coefficient matrices and righthandsides. This algorithm can reduce the number of variables. After that, we present the algorithm incSolver (for incremental system solver) which utilizes similarity in the iterations of the program for a linear equation system. All three techniques can be used in column generation of multicommodity problems. Preliminary numerical experiments show that the incSolver is significantly faster than the existing algorithms. For example, random test cases show that incSolver is at least 37 times and up to 341 times faster than popular solver LAPACK.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.07469
 Bibcode:
 2019arXiv190307469D
 Keywords:

 Mathematics  Optimization and Control
 EPrint:
 27 pages