Optimization and Engineering

, Volume 14, Issue 4, pp 519–527 | Cite as

On sparse matrix orderings in interior point methods

  • Csaba MészárosEmail author


The major computational task of most interior point implementations is solving systems of equations with symmetric coefficient matrix by direct factorization methods, therefore, the performance of Cholesky-like factorizations is a critical issue. In the case of sparse and large problems the efficiency of the factorizations is closely related to the exploitation of the nonzero structure of the problem. A number of techniques were developed for fill-reducing sparse matrix orderings which make Cholesky factorizations more efficient by reducing the necessary floating point computations. We present a variant of the nested dissection algorithm incorporating special techniques that are beneficial for graph partitioning problems arising in the ordering step of interior point implementations. We illustrate the behavior of our algorithm and provide numerical results and comparisons with other sparse matrix ordering algorithms.


Sparse matrix ordering Nested dissection Interior point methods 


  1. Andersen ED, Gondzio J, Mészáros C, Xu X (1996) Implementation of interior point methods for large scale linear programs. In: Terlaky T (ed) Interior point methods of mathematical programming. Kluwer Academic, Norwel, pp 189–252 CrossRefGoogle Scholar
  2. Ashcraft C, Liu JWH (1997) Using domain decomposition to find graph bisectors. BIT Numer Math 37(3):506–534 MathSciNetCrossRefzbMATHGoogle Scholar
  3. Ashcraft C, Liu JWH (1998a) Applications of the Dulmage–Mendelsohn decomposition and network ow to graph bisection improvement. SIAM J Matrix Anal Appl 19:325–354 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Ashcraft C, Liu JWH (1998b) Robust ordering of sparse matrices using multisection. SIAM J Matrix Anal Appl 19(3):816–832 MathSciNetCrossRefzbMATHGoogle Scholar
  5. Booth KS, Lipton RJ (1981) Computing extremal and approximate distances in graphs having unit cost edges. Acta Inform 15(4):319–328 MathSciNetzbMATHGoogle Scholar
  6. Duff IS, Erisman AM, Reid JK (1986) Direct methods for sparse matrices. Oxford University Press, New York zbMATHGoogle Scholar
  7. Fiduccia C, Mattheyses R (1982) A linear-time heuristic for improving network partition. Technical report, ACM IEEE 19th design and automation conference proceedings Google Scholar
  8. George JA (1973) Nested dissection of a regular finite element mesh. SIAM J Numer Anal 10:345–363 MathSciNetCrossRefzbMATHGoogle Scholar
  9. George A, Liu JWH (1981) Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood Cliffs zbMATHGoogle Scholar
  10. George A, Liu JWH (1989) The evolution of the minimum degree ordering algorithm. SIAM Rev 31:1–19 MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gupta A (1996) Watson graph partitioning package. Technical report RC 20453, IBM T. J. Watson Research Center Google Scholar
  12. Hendrickson B, Leland R (1995) The Chaco user’s guide, version 2.0. Technical report, Sandia National Laboratories Google Scholar
  13. Hendrickson B, Rothberg E (1998) Improving the run time and quality of nested dissection ordering. SIAM J Sci Comput 20(2):468–489 MathSciNetCrossRefGoogle Scholar
  14. Karypis G, Kumar V (1995) Analysis of multilevel graph partitioning. Technical report, Supercomputing ’95, proceedings of the 1995 ACM/IEEE conference on supercomputing Google Scholar
  15. Karypis G, Kumar V (1998) Metis a software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices version 3.0. Technical report, University of Minnesota Google Scholar
  16. Karypis G, Kumar V (1999) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392 MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kernighan BW, Lin S (1970) An efficient heuristic procedure for partitioning graphs. Technical report 49, Bell Syst Tech J Google Scholar
  18. Lustig IJ, Marsten RE, Shanno DF (1994) Interior point methods for linear programming: computational state of the art. ORSA J Comput 6(1):1–15 MathSciNetCrossRefzbMATHGoogle Scholar
  19. Mészáros C (1997) The augmented system variant of IPMs in two-stage stochastic linear programming. Eur J Oper Res 101(2):317–327 CrossRefzbMATHGoogle Scholar
  20. Mészáros C (1998) Ordering heuristics in interior point LP methods. In: Gianessi F, Komlósi S, Rapcsák T (eds) New trends in mathematical programming. Kluwer Academic, Norwel, pp 203–221 CrossRefGoogle Scholar
  21. Mészáros C (1999) The BPMPD interior-point solver for convex quadratic problems. Optim Methods Softw 11/12:431–449 CrossRefGoogle Scholar
  22. Mészáros C (2007) Detecting dense columns in linear programs for interior point methods. Comput Optim Appl 36(2–3):309–320 MathSciNetCrossRefzbMATHGoogle Scholar
  23. Mészáros C (2010) On the implementation of interior point methods for dual-core platforms. Optim Methods Softw 25(3):449–456 MathSciNetCrossRefzbMATHGoogle Scholar
  24. Mészáros C, Suhl UH (2004) Advanced preprocessing techniques for linear and quadratic programming. OR Spektrum 25:575–595 CrossRefGoogle Scholar
  25. Mittelmann HD, Spellucci P (1998) Decision tree for optimization software. World Wide Web.
  26. Parter SV (1961) The use of linear graphs in Gaussian elimination. SIAM Rev 3:130–191 MathSciNetGoogle Scholar
  27. Rothberg E (1996) Ordering sparse matrices using approximate minimum local fill. Technical report, Silicon Graphics Inc., Mountain View, CA 94043 Google Scholar
  28. Rothberg E, Hendrickson B (1998) Sparse matrix ordering methods for interior point linear programming. INFORMS J Comput 10(1):107–113 MathSciNetCrossRefGoogle Scholar
  29. Yannakakis M (1981) Computing the minimum fill-in is NP-complete. SIAM J Algebr Discrete Methods 2:77–79 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Group of Operations Research and Decision SystemsHungarian Academy of SciencesBudapestHungary

Personalised recommendations