Abstract
We devise a hybrid approach for solving linear systems arising from interior point methods applied to linear programming problems. These systems are solved by preconditioned conjugate gradient method that works in two phases. During phase I it uses a kind of incomplete Cholesky preconditioner such that fill-in can be controlled in terms of available memory. As the optimal solution of the problem is approached, the linear systems becomes highly ill-conditioned and the method changes to phase II. In this phase a preconditioner based on the LU factorization is found to work better near a solution of the LP problem. The numerical experiments reveal that the iterative hybrid approach works better than Cholesky factorization on some classes of large-scale problems.
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Adler, I., Karmarkar, N.K., Resende, M.G.C., Veiga, G.: Data structures and programming techniques for the implementation of Karmarkar’s algorithm. ORSA J. Comput. 1(2), 84–106 (1989)
Adler, I., Karmarkar, N.K., Resende, M.G.C., Veiga, G.: An implementation of Karmarkar’s algorithms for linear programming. Math. Program. 44, 297–335 (1989)
Altman, A., Gondzio, J.: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optim. Methods Software 11, 275–302 (1999)
As̆ić, M.D., Kovac̆ević-Vujc̆ić, V.V.: Ill-conditionedness and interior-point methods. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11, 53–58 (2000)
Baryamureeba, V.: Solution of large-scale weighted least squares problems. Numer. Linear Algebra Appl. 9, 93–106 (2002)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28(2), 149–171 (2004)
Bunch, J.R., Parlett, B.N.: Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639–655 (1971)
Campos, F.F.: Analysis of conjugate gradients—type methods for solving linear equations. Ph.D. thesis, Oxford University Computing Laboratory, Oxford (1995)
Campos, 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. SIAM J. Sci. Comput. 19(1), 126–138 (1998)
Czyzyk, J., Mehrotra, S., Wagner, M., Wright, S.J.: PCx an interior point code for linear programming. Optim. Methods Software 11(2), 397–430 (1999)
Dollar, H.S., Gould, N.I.M., Wathen A.J.: On implicit-factorization constraint preconditioners. Technical Report RAL-TR-2004-036, Rutherford Appleton Laboratory. Also In: Di Pillo, G., Roma, M., (eds.) Large Scale Nonlinear Optim. Springer (2004, to appear)
Duff, I.S., Grimes, R.G., Lewis, J.G.: User’s guide for the Harwell–Boeing sparse matrix collection (release I). Technical Report PA-92-86, CERFACS, Toulouse, France (1992)
Durazzi, C., Ruggiero, V.: Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems. Numer. Linear Algebra Appl. 10, 673–688 (2003)
Forsythe, G.E., Straus, E.G.: On best conditioned matrices. Proc. Am. Math. Soc. 6, 340–345 (1955)
George, A., Ng, E.: An implementation of Gaussian elimination with partial pivoting for sparse systems. SIAM J. Sci. Stat. Comput. 6, 390–409 (1985)
Gill, P.E., Murray, W., Ponceléon, D.B., Saunders, M.A.: Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl. 13, 292–311 (1992)
Gondzio, J.: HOPDM (Version 2.12)—A fast LP solver based on a primal–dual interior point method. Eur. J. Oper. Res. 85, 221–225 (1995)
Gondzio, J.: Multiple centrality corrections in a primal–dual method for linear programming. Comput. Optim. Appl. 6, 137–156 (1996)
Gould, N.I.M., Hribar, M.E., Nocedal, J.: On the solution of equality constrained quadratic programming problems arising in optimization. SIAM J. Sci. Comput. 23, 1376–1395 (2001)
Haws, J.C., Meyer, C.D.: Preconditioning KKT systems. Online at http://meyer.math.ncsu.edu/Meyer/PS_Files/KKT.pdf
Hungarian Academy of Sciences OR Lab: Miscellaneous LP models. Online at http://www.sztaki.hu/~meszaros/public_ftp/lptestset/misc
Hungarian Academy of Sciences OR Lab: Stochastic LP test sets. Online at http://www.sztaki.hu/~meszaros/public_ftp/lptestset/stochlp
Jones, M.T., Plassmann, P.E.: An improved incomplete Cholesky factorization. ACM Trans. Math. Software 21, 5–17 (1995)
Karisch, S., Burkard, R.S., Rendl, F.: QAPLIB—a quadratic assignment problem library. Eur. J. Oper. Res. 55, 115–119 (1991)
Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21(4), 1300–1317 (2000)
Lustig, I.J., Marsten, R.E., Shanno, D.F.: On implementing Mehrotra’s predictor-corrector interior point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)
Manteuffel, T.A.: An incomplete factorization technique for positive definite linear systems. Math. Comput. 34(150), 473–497 (1980)
Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)
Mittelmann–LP models. Miscellaneous LP models collect by Hans D. Mittelmann. Online at ftp://plato.asu.edu/pub/lptestset/pds
Munksgaard, N.: Solving sparse symmetric sets of linear equations by preconditioned conjugate gradients. ACM Trans. Math. Software 6(2), 206–219 (1980)
NETLIB LP repository: NETLIB collection LP test sets. Online at http://www.netlib.org/lp/data
Oliveira, A.R.L.: A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, Houston (1997)
Oliveira, A.R.L., Sorensen, D.C.: A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Linear Algebra Appl. 394, 1–24 (2005)
Padberg, M., Rijal, M.P.: Location, Scheduling, Design and Integer Programming. Kluwer Academic, Boston (1996)
Portugal, L.F., Resende, M.G.C., Veiga, G., Júdice, J.J.: A truncated primal-infeasible dual-feasible network interior point method. Networks 35, 91–108 (2000)
Resende, M.G.C., Veiga, G.: An implementation of the dual affine scaling algorithm for minimum cost flow on bipartite uncapaciated networks. SIAM J. Optim. 3, 516–537 (1993)
SLATEC collection. The dsics.f subroutine in NETLIB repository. Online at http://www.netlib.org/slatec/lin/dsics.f
Wang, W., O’Leary, D.P.: Adaptive use of iterative methods in predictor-corrector interior point methods for linear programming. Numer. Algorithms 25(1–4), 387–406 (2000)
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Bocanegra, S., Campos, F.F. & Oliveira, A.R.L. Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods. Comput Optim Appl 36, 149–164 (2007). https://doi.org/10.1007/s10589-006-9009-5
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DOI: https://doi.org/10.1007/s10589-006-9009-5