Solving Quadratic Multicommodity Problems through an Interior-Point Algorithm

Abstract

Standard interior-point algorithms usually show a poor performance when applied to multicommodity network flow problems. A recent specialized interior-point algorithm for linear multicommodity network flows overcame this drawback, and was able to efficiently solve large and difficult instances. In this work we perform a computational evaluation of an extension of that specialized algorithm for multicommodity problems with convex and separable quadratic objective functions. As in the linear case, the specialized method for convex separable quadratic problems is based on the solution of the positive definite system that appears at each interior-point iteration through a scheme that combines direct (Cholesky) and iterative (preconditioned conjugate gradient) solvers. The preconditioner considered for linear problems, which was instrumental in the performance of the method, has shown to be even more efficient for quadratic problems. The specialized interior-point algorithm is compared with the general barrier solver of CPLEX 6.5, and with the specialized codes PPRN and ACCPM, using a set of convex separable quadratic multicommodity instances of up to 500000 variables and 180000 constraints. The specialized interior-point method was, in average, about 10 times and two orders of magnitude faster than the CPLEX 6.5 barrier solver and the other two codes, respectively.