A Study of Preconditioners for Network Interior Point Methods
Computational Optimization and Applications
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Recently, Resende and Veiga [{\it SIAM J.\ Optim.}, 3 (1993), pp. 516--537] proposed an efficient implementation of the dual affine (DA) interior-point algorithm for the solution of linear transportation models with integer costs and right-hand-side coefficients. This procedure incorporates a preconditioned conjugate gradient (PCG) method for solving the linear system that is required in each iteration of the DA algorithm. In this paper, we introduce an incomplete $QR$ decomposition (IQRD) preconditioning for the PCG algorithm. Computational experience shows that the IQRD preconditioning is appropriate in this instance and is more efficient than the preconditioning introduced by Resende and Veiga. We also show that the primal dual (PD) and the predictor corrector (PC) interior-point algorithms can also be implemented by using the same type of technique. A comparison among these three algorithms is included and indicates that the PD and PC algorithms are more appropriate for the solution of transportation problems with well-scaled cost and right-hand-side coefficients and assignment problems with poorly scaled cost coefficients. On the other hand, the DA algorithm seems to be more efficient for assignment problems with well-scaled cost coefficients and transportation problems whose cost coefficients are badly scaled.