Skip to main content
SpringerLink
Log in

Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

Quadratic programming, symmetry, positive (semi) definiteness and the linear complementary problem were generalized by Morris and Todd to oriented matroids. Todd gave a constructive solution for the quadratic programming problem of oriented matroids. Using Las Vergnas' lexicographic extension and Bland's basic tableau construction Todd generalized Lemke's quadratic programming algorithm for this problem.

Here some generalizations of Terlaky's finite criss-cross method are presented for oriented matroid quadratic programming. These algorithms are based on the smallest subscript rule and on sign patterns, and do not preserve feasibility on any subsets. In fact two variants of the generalized criss-cross method are presented. Finally two special cases (oriented matroid linear programming and the definite case) are discussed.

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

Access this article

Log in via an institution

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. M. L. Balinski andA. W. Tucker, Duality theory of linear programs: A constructive approach with applications,SIAM Review,11 (1969), 347–377.

    Google Scholar 

  2. E. M. L. Beale, On quadratic programming,Naval Research Logistics Quarterly,6 (1959), 227–244.

    Google Scholar 

  3. M. J. Best, Equivalence of some quadratic programming algorithms,Mathematical Programming,30 (1984), 71–87.

    Google Scholar 

  4. R. G. Bland, New finite pivoting rules for the simplex method,Mathematics of Operations Research,2 (1977), 103–107.

    Google Scholar 

  5. R. G. Bland, A combinatorial abstraction of linear programming,J. Combinatorial Theory Ser. B.,23 (1977), 33–57.

    Google Scholar 

  6. R. G. Bland andM. Las Vergnas, Orientability of matroids,J. Combinatorial Theory Ser. B.,24 (1978), 94–123.

    Google Scholar 

  7. Y. Y. Chang andR. W. Cottle, Least index resolution of degeneracy in quadratic programming,Mathematical Programming,18 (1980), 127–137.

    Google Scholar 

  8. R. W. Cottle, The principal pivoting method of quadratic programming, in:Mathematics of the Decision Sciences Part I. Eds. G. B. Dantzig and A. F. Veinott, Lectures in Applied Mathematics, 11 (American Mathematical Society, Providence, R. I., (1968), 142–162.

    Google Scholar 

  9. R. W. Cottle andG. B. Dantzig, Complementary pivot theory of mathematical programming,Linear Algebra and its Applications,1 (1986), 103–125.

    Google Scholar 

  10. G. B. Dantzig,Linear programming and Extensions, Princeton University Press, Princeton, N. J., 1963.

    Google Scholar 

  11. J. Folkman andJ. Lawrence, Oriented matroids,J. Combinatorial Theory Ser. B.,25 (1978), 199–236.

    Google Scholar 

  12. K.Fukuda, Oriented matroid programming,Ph. D. dissertation, University of Waterloo, 1982.

  13. D.Jensen, Coloring and Duality: Combinatorial Augmentation methods,Ph. D. dissertation School of OR and IE, Cornell University, 1985.

  14. E. Keller, The general quadratic optimization problem,Mathematical Programming,5 (1973), 311–337.

    Google Scholar 

  15. E.Klafszky and T.Terlaky, Variants of the „Hungarian Method” for solving linear programming problems,Optimization (accepted for publication).

  16. E.Klafszky and T.Terlaky, Some generalizations of the criss-cross method for quadratic programming,Mathematics of Operations Research (to appear).

  17. M. Las Vergnas, Bases in oriented matroids,J. Combinatorial Theory Ser. B.,25 (1978), 283–289.

    Google Scholar 

  18. M. Las Vergnas, Convexity in oriented matroids,J. Combinatorial Theory Ser. B.,29 (1980), 231–243.

    Google Scholar 

  19. J. Lawrence, Oriented matroids and multiply ordered sets,Linear Algebra and its Applications,48 (1982), 1–12.

    Google Scholar 

  20. C. E. Lemke, Bimatrix equilibrium points and mathematical programming,Management Science,11 (1965), 681–689.

    Google Scholar 

  21. W. D.Morris, Oriented matroids and the linear complementarity problem,Ph. D. thesis. School of OR and IE, Cornell University, 1986.

  22. W. D.Morris and M. J.Todd, Symmetry and positive definiteness in oriented matroids,Cornell University, School of OR and IE, Itacha, Itacha N. Y. Technical Report No.631.

  23. R. T. Rockafellar, The elementary vectors of a subspace ofR n, inCombinatorial Mathematics and its Applications, Proc. of the Chapel Hill Conference, 1967 (R. G. Bore and T. A. Dowling, Eds.) 104–127, Univ. of North Carolina Press, Chapel Hill. 1969.

    Google Scholar 

  24. K.Ritter, A dual quadratic programming algorithm,Univ. of Wisconsin-Madison, Mathematics Research Center, Technical Summary Report, No. 2733 (1984).

  25. C.Roos, On the Terlaky path in the umbrella graph of a linear programming problem,Reports of the Department of Informatics, Delft University of Technology, No. 85-12.

  26. T. Terlaky, A convergent criss-cross method,Optimization,16 (1985), 5, 683–690.

    Google Scholar 

  27. T. Terlaky, A finite criss-cross method for oriented matroids,J. Combinatorial Theory Ser. B.,42 (1987), 319–327.

    Google Scholar 

  28. T. Terlaky, A new algorithm for quadratic programming,European Journal of Operations Research,32 (1987), 294–301.

    Google Scholar 

  29. T. Terlaky, Onlp programming,European Journal of Operations Research,22 (1985), 70–101.

    Google Scholar 

  30. M. J. Todd, Complementarity in oriented matroids,SIAM J. Alg. Disc. Math.,5 (1984), 467–485.

    Google Scholar 

  31. M. J. Todd, Linear and quadratic programming in oriented matroids,J. Combinatorial Theory Ser. B.,39 (1985), 105–133.

    Google Scholar 

  32. A. W. Tucker, Principal pivotal transforms of square matrices,SIAM Review,5 (1963), 305.

    Google Scholar 

  33. C. Van de Panne andA. Whinston, The symmetric formulation of the simplex method for quadratic programming,Econometrica,37 (1969), 507–527.

    Google Scholar 

  34. D. A.Welsh, Matroid theory,Academic Press, 1976.

  35. P. Wolfe, The simplex method for quadratic programming,Econometrica,27 (1959), 382–398.

    Google Scholar 

  36. Zh. Wang, A finite conformal-elimination-free algorithm for oriented matroid programming,Chinecse Annals of Mathematics,8B. No. 1 (1987).

  37. H. Whitney, On the abstract properties of linear dependence,Amer. J. Math.,57 (1935), 507–553.

    Google Scholar 

  38. S. Zionts, The criss-cross method for solving linear programming problems,Management Science,15 (1969), 426–445.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Technical University, H-35I5, Miskolc, Miskolc-Egyetemváros, Hungary

    E. Klafszky

  2. Dept. of Op. Res., Eötvös University, Múzeum krt. 62-8, H-1088, Budapest, Hungary

    T. Terlaky

Authors
  1. E. Klafszky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. Terlaky
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Klafszky, E., Terlaky, T. Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids. Combinatorica 9, 189–198 (1989). https://doi.org/10.1007/BF02124679

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02124679

AMS subject classification (1980)

Search

Navigation