Skip to main content
Springer Nature Link
Log in

A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming

  • Published:
Mathematical Programming Submit manuscript

Abstract.

Cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods (IPMs) for linear programming (LP), its main advantages being numerical stability and efficiency for sparse systems. However in the presence of dense columns there is dramatic loss of efficiency. A typical approach to remedy this problem is to apply the Sherman-Morrison-Woodbury (SMW) update formula to the dense part. This approach while being very efficient, is not numerically stable. Here we propose using product-form Cholesky factorization to handle dense columns. The proposed approach is essentially as stable as the original Cholesky factorization and nearly as efficient as the SMW approach. We demonstrate these properties both theoretically and computationally. A key part of our theoretical analysis is the proof that the elements of the Cholesky factors of the matrices that arise in IPMs for LP are uniformly bounded as the duality gap converges to zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, I., Karmarkar, N., Resende, M.G.C., Veiga, G.: Data structures and programming techniques for implementation of Karmarkar’s algorithm. ORSA J. on Computing 1(2), 84–106 (1989)

    MATH  Google Scholar 

  2. Andersen, E.D., Gondzio, J., Mészáros, C., Xu, X.: Implementation of interior point methods for large-scale linear programming. In: Interior Point Methods of Mathematical Programming. Tamás Terlaky (Ed.) Kluwer Academic Publishers, 1996

  3. Andersen, K.D.: A modifies Schur complement method for handling dense columns in interior point methods for linear programming. ACM Transactions on Mathematical Software 22(3), 348–356 (1996)

    Article  MATH  Google Scholar 

  4. Bennet, J.M.: Triangular factors of modified matrices. Numerisches Mathematik 7, 217–221 (1965)

    Google Scholar 

  5. Choi, I.C., Monma, C.L., Shanno, D.F.: Further development of primal-dual interior point methods. ORSA J. on Computing 2(4), 304–311 (1990)

    MATH  Google Scholar 

  6. Dikin, I.I.: On the speed of an iterative process. Upravlaemye Sistemy 12, 54–60 (1974)

    MATH  Google Scholar 

  7. Fletcher, R., Powell, M.J.D.: On the Modification of LDL t factorization. Mathematics of Computation 28(128), 1067–1087 (1974)

    MATH  Google Scholar 

  8. Gay, D.M.: Electronic mail distribution of linear programming test problems. Mathematical Programming Society COAL Newsletter 13, 10–12 (1985)

    Google Scholar 

  9. Fourer, R., Mehrotra, S.: Solving symmetric indefinite systems in an interior point method for linear programming. Mathematical Programming 62, 15–40 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Gill, Ph.E., Murray, W., Saunders, M.A.: Methods for computing and modifying the LDV factors of a matrix. Mathematics of Computation 29, 1051–10 (1975)

    MATH  Google Scholar 

  11. Gill, Ph.E., Golub, G.H., Murray, W., Saunders, M.A.: Methods for modifying matrix factorizations. Mathematics of Computation 28, 505–535 (1974)

    MATH  Google Scholar 

  12. Gondzio, J.: Splitting dense columns of constraint matrix in interior point methods for large scale linear programming. Optimization 24, 285–297 (1992)

    MathSciNet  MATH  Google Scholar 

  13. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM publications, Philadelphia, 1996

  14. Hough, P.: Stable computation of search directions for near-degenerate linear programming problems. Sandia Report, 97-8243

  15. Karmarkar, N.K.: A polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    MathSciNet  MATH  Google Scholar 

  16. Lustig, I.L., Marsten, R.E., Shanno, D.F.: Interior point methods for linear programming: Computational state of the art. ORSA J. on Computing 6(1), 1–15 (1994)

    Google Scholar 

  17. Maros, I., Mészáros, Cs.: The role of the augmented system in the interior point methods. European Journal of Operationsl Research 107, 720–736 (1998)

    Article  MATH  Google Scholar 

  18. Marxen, A.: Primal barrier methods for linear programming. Report SOL 89-6, Dept. of Operations Research, Stanford University, Stanford, CA, 1989

  19. McShane, K.A., Monma, C.L., Shanno, D.F.: An implementation of a primal-dual method for linear programming. ORSA J. on Computing 1(2), 70–83 (1989)

    MATH  Google Scholar 

  20. Megiddo, N.: Pathways to the optimal set in linear programming. In: Progress in Mathematical Programming: Interior-Point Algorithms and Related Methods, N. Megiddo, ed., Springer, Berlin, 1989, pp. 131–158

  21. Mehrotra, S.: Handling free variables in interior methods. Technical Report 91-06, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, USA, 1991

  22. Mehrotra, S.: On implementation of a primal-dual interior point method. SIAM J. on Optim. 2(4), 575–601 (1992)

    MATH  Google Scholar 

  23. Mészáros, Cs.: The augmented system variant of IPMs in two-stage stochastic linear programming computation. Working paper WP 95-11, Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, 1995

  24. Portugal, L., Bastos, F., Judice, J., Paixõ, J., Terlaky, T.: An investigation of interior point algorithms for the linear transportation problems. SIAM Journal on Scientific Computing 17(5), 1202–1223 (1996)

    Article  MATH  Google Scholar 

  25. Resende, M.G.C., Veiga, G.: An efficient implementation of a network interior point method. Network Flows and Matching: First DIMACS Implementation Challenge, D.S. Johnson, C.C. McGeoch, eds., DIMACS Series on Discrete Mathematics and Theoretical Computer Science, 12 1993, pp. 299–348

  26. Stewart, G.W.: On scaled projections and pseudoinverses. Linear Algebra and Applications 112, 189–193 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Todd, M.J.: A Dantzig-Wolfe-like variant of Karmarkar’s interior-point linear programming algorithm. Operations Research 38, 1006–1018 (1990)

    MATH  Google Scholar 

  28. Vanderbei, R.J.: Splitting dense columns in sparse linear systems. Linear Algebra and Applications 152, 107–117 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vavasis, S.A., Ye, Y.: Condition numbers for polyhedra with real number data. Operations Research Letters 17, 209–214 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice-Hall, Englewood Cliffs, NJ. 1963

  31. Wright, S.J.: Stability of linear equations solvers in interior-point methods. SIAM J. on Matrix Analysis and Applications 16, 1287–1307 (1994)

    MATH  Google Scholar 

  32. Wright, S.J.: Modified Cholesky factorizations in interior point algorithms for linear programming. SIAM J. on Optimization 9(4), 1159–1191 (1999)

    Article  MATH  Google Scholar 

  33. Wright, S.J.: Primal-Dual Interior Point Methods, SIAM, Philadelphia, 1997

Download references

Author information

Authors and Affiliations

  1. Dept. of IEOR, Columbia University, New York, NY

    D. Goldfarb

  2. IBM, T.J. Watson Research Center, Yorktown Heights, NY

    K. Scheinberg

Authors
  1. D. Goldfarb
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. K. Scheinberg
    View author publications

    You can also search for this author inPubMed Google Scholar

Additional information

The doctoral research of this author was supported in part by an IBM Cooperative Fellowship

Research supported in part by NSF Grants DMS 91-06195, DMS 94-14438, DMS 95-27124, DMS 01-04282 and CDA 97-26385 and DOE Grant DE-FG02-92ER25126

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldfarb, D., Scheinberg, K. A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming. Math. Program., Ser. A 99, 1–34 (2004). https://doi.org/10.1007/s10107-003-0377-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0377-7

Keywords