Heuristics For Implementation Of A Hybrid Preconditioner For Interior-point Methods

This article presents improvements to the hybrid preconditioner previously developed for the solution through the conjugate gradient method of the linear systems which arise from interior-point methods. The hybrid preconditioner consists of combining two preconditioners: controlled Cholesky factorization and the splitting preconditioner used in different phases of the optimization process. The first, with controlled fill-in, is more efficient at the initial iterations of the interior-point methods and it may be inefficient near a solution of the linear problem when the system is highly ill-conditioned; the second is specialized for such situation and has the opposite behavior. This approach works better than direct methods for some classes of large-scale problems. This work has proposed new heuristics for the integration of both preconditioners, identifying a new change of phases with computational results superior to the ones previously published. Moreover, the performance of the splitting preconditioner has been improved through new orderings of the constraint matrix columns allowing savings in the preconditioned conjugate gradient method iterations number. Experiments are performed with a set of large-scale problems and both approaches are compared with respect to the number of iterations and running time. © 2011 Bralizian Operations Research Society.313579591Adler, I., Karmarkar, N., Resende, M.G.C., Veiga, G., Data structures and programming techniques for the implementation of Karmarkar's algorithm (1989) ORSA Journal on Computing, pp. 84-106Bergamaschi, L., Gondzio, J., Zilli, G., Preconditioning Indefinite Systems in Interior Point Methods for Optimization (2004) Computational Optimization and Applications, 28, pp. 149-171Bocanegra, S., Campos, F.F., Oliveira, A.R.L., Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods (2007) Computational Optimization and Applications, (1-2), pp. 149-164. , Specialissueon"LinearAlgebraIssuesarisinginInteriorPointMethods"Burkard, R.S., Karisch, S., Rendl, F., QAPLIB-A quadratic assignment problem library (1991) European Journal of Operations Research, 55, pp. 115-119Campos, F.F., Birkett, N.R.C., An efficient solver for multi-right-hand-side linear systems based on the CCCG(η) method with applications to implicit time-dependent partial differential equations (1998) SIAM J Sci Comput, 19 (1), pp. 126-138Czyzyk, J., Mehrotra, S., Wagner, M., Wright, S.J., PCx an interior point code for linear programming (1999) Optimization Methods and Software, 11 (2), pp. 397-430Golub, G.H., van Loan, C.F., (1996) Matrix Computations, , Third Edition, The Johns Hopkins University Press, BaltimoreJones, M.T., Plassmann, P.E., An improved Incomplete Cholesky Factorization (1995) ACM Trans on Math Software, 21 (1), pp. 5-17Lustig, I.J., Marsten, R.E., Shanno, D.F., On Implementing Mehrotra's Predictor-Corrector Interior Point Method for Linear Programming (1990) TR SOR, pp. 03-90Mehrotra, S., On the Implementation of a Primal-Dual Interior Point Method (1992) SIAM Journal on Optimization, 2 (4), pp. 575-601Mehrotra, S., Implementations of Affine Scaling Methods: Approximate Solutions of Systems of Linear Equations Using Preconditioned Conjugate Gradient Methods (1992) ORSA Journal on Computing, 4 (2), pp. 103-118Monteiro, R.D.C., Ilan, A.D.L.E.R., Resende, M.G.C., A Polynomial-time Primal-Dual Affine Scaling Algorithm for Linear and Convex Quadratic Programming and its Power Series Extension (1990) Mathematics of Operations Research, 15, pp. 191-214Oliveira, A.R.L., Sorensen, D.C., A new class of preconditioners for large-scale linear systems from interior point methods for linear programming (2005) Linear Algebra and its applications, 394, pp. 1-24Padberg, M., Rijal, M.P., (1996) Location, Scheduling, Design and Integer Programming, , Kluwer Academic, BostonWang, W., O'Leary, D.P., Adaptive use of iterative methods in predictor-corrector interior point methods for linear programming (2000) Numerical Algorithms, 25, pp. 387-406Wright, S.J., (1996) Primal-Dual Interior-Point Methods, , SIAM Publications, SIAM, Philadelphia, PA, US