Skip to main content
SpringerLink
Log in

Degeneracy in linear complementarity problems: a survey

  • Part I: Surveys
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Given a square matrixM of ordern and a vectorq ∈ ℝn, the linear complementarity problem is the problem of either finding aw ∈ ℝn and az ∈ ℝn such thatw−Mz=q,w≥0,z≥0 andw T z=0 or showing that no such (w, z) exists. This problem is denoted asLCP(q, M). We say that a solution (w, z) toLCP(q, M) is degenerate if the number of positive coordinates in (w, z) is less thann. As in linear programming, degeneracy may cause cycling in an adjacent vertex following methods like Lemke's algorithm. Moreover, ifLCP(0,M) has a nontrivial solution, a condition related to degeneracy, then unless certain other conditions are satisfied, the algorithm may not be able to decide about the solvability of the givenLCP(q, M). In this paper we review the literature on the implications of degeneracy to the linear complementarity theory.

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. Aganagic and R.W. Cottle, A note onQ-matrices, Math. Progr. 16 (1979) 374.

    Article  Google Scholar 

  2. M.S. Bazaraa and C.M. Shetty,Nonlinear Programming — Theory and Algorithms (Wiley, New York, 1979).

    Google Scholar 

  3. R.G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2 (1977) 103.

    Article  Google Scholar 

  4. R. Chandrasekaran, A special case of the complementary pivot algorithm, Opsearch 7 (1970) 263.

    Google Scholar 

  5. Y.Y. Chang, Least index resolution of degeneracy in linear complementarity problems, Technical Report 79-14, Department of Operations Research, Stanford University (1979).

  6. R.W. Cottle and Y.Y. Chang, Least index resolution of degeneracy in linear complementarity problems with sufficient matrices, Technical Report SOL 90-9, Systems Optimization Laboratory, Department of Operations Research, Stanford University (1990).

  7. R.W. Cottle and G.B. Dantzig, Complementary pivot theory of mathematical programming, Lin. Alg. Appl. 1 (1968) 103.

    Article  Google Scholar 

  8. R.W. Cottle, Solution rays for a class of complementarity problems, Math. Progr. Study 1 (1974) 59.

    Google Scholar 

  9. R.W. Cottle, CompletelyQ-matrices, Math. Progr. 19 (1980) 347.

    Article  Google Scholar 

  10. R.W. Cottle and R.E. Stone, On the uniqueness of solutions to linear complementarity problems, Math. Progr. 27 (1983) 191.

    Article  Google Scholar 

  11. C.W. Cryer, The method of Christopherson for solving the free boundary problems for infinite journal bearing by means of finite differences, Math. Comp. 25 (1971) 435.

    Article  Google Scholar 

  12. R.D. Doverspike, Some perturbation results for the linear complementarity problems, Math. Progr. 23 (1982) 181.

    Article  Google Scholar 

  13. N. Eagambaram and S.R. Mohan, On strongly degenerate complementary cones and solution rays, Math. Progr. 44 (1989) 77.

    Article  Google Scholar 

  14. N. Eagambaram and S.R. Mohan, On some classes of linear complementarity problems with matrices of ordern and rank (n−1), Math. Oper. Res. 15 (1990) 243.

    Article  Google Scholar 

  15. N. Eagambaram and S.R. Mohan, A note on the linear complementarity problem with anN 0-matrix, Arabian J. Sci. Eng. 16 (1991) 341.

    Google Scholar 

  16. B.C. Eaves, The linear complementarity problem, Manag. Sci. 17 (1971) 612.

    Article  Google Scholar 

  17. C.B. Garcia, Some classes of matrices in linear complementarity theory, Math. Progr. 5 (1973) 299.

    Article  Google Scholar 

  18. M.S. Gowda, On the continuity of the solution map of the linear complementarity problem, Research Report, 90-23, University of Maryland at Baltimore County (1990).

  19. G. Hadley,Linear Programming (Addison-Wesley, Reading, MA, 1962).

    Google Scholar 

  20. M.W. Jeter and W.C. Pye, An example of nonregular semi-monotoneQ-matrix, Math. Progr. 44 (1989) 331

    Article  Google Scholar 

  21. I. Kaplansky, A contribution to Von Neumann's theory of games, Ann. Math. 146 (1945) 474.

    Article  Google Scholar 

  22. S. Karamardian, The complementarity problem, Math. Progr. 2 (1972) 107.

    Article  Google Scholar 

  23. M. Kostreva, Cycling in linear complementarity problems, Math. Progr. 16 (1979) 127.

    Article  Google Scholar 

  24. L.C. Larsen,Problem-Solving through Problems (Springer, 1983).

  25. C.E. Lemke and J.T. Howson Jr., Equilibrium points of bimatrix games, SIAM J. Appl. Math. 12 (1964) 413.

    Article  Google Scholar 

  26. C.E. Lemke, Bimatrix equilibrium points and mathematical programming, Manag. Sci. 11 (1965) 681.

    Article  Google Scholar 

  27. G. Maier, A matrix structural theory of linear elastoplasticity with interacting yield planes, Meccanica 5 (1970) 45.

    Google Scholar 

  28. G. Maier, Problem 72-7, A parametric linear complementary problem, SIAM Rev. 14 (1972) 364.

    Article  Google Scholar 

  29. O.L. Mangasarian, Linear complementarity problems solvable by a single linear program, Math. Progr. 10 (1976) 263.

    Article  Google Scholar 

  30. K.J. Marshall and J.W. Suurballe, A note on cycling in the simplex method, Naval Res. Log. Quarterly 16 (1969) 121.

    Google Scholar 

  31. R.J. McElicea,The Theory of Information and Coding, Encyclopedia of Mathematics (Addison-Wesley, Reading, MA, 1973).

    Google Scholar 

  32. S.R. Mohan, T. Parthasarathy and R. Sridhar,\(\bar N\)-matrices and the classQ, Technical Report No. 9110, Indian Statistical Institute, Delhi Centre (1991).

  33. K.G. Murty, On the number of solutions to the complementarity problem and spanning properties of complementary cones, Lin. Alg. Appl. 5 (1972) 65.

    Article  Google Scholar 

  34. K.G. Murty,Linear Programming (Wiley, New York, 1983).

    Google Scholar 

  35. K.G. Murty,Linear Complementarity, Linear and Non-linear Programming (Heldermann, Berlin, 1988).

    Google Scholar 

  36. W.C. Pye, AlmostP 0-matrices and the classQ, Technical Report, Department of Mathematics, University of Southern Mississippi (June 1990).

  37. R. Saigal, A characterization of the constant parity property of the number of solutions to the linear complementarity problem, SIAM J. Appl. Math. 23 (1972) 40.

    Article  Google Scholar 

  38. R. Saigal, On the class of complementary cones and Lemke's algorithm, SIAM J. Appl. Math. 23 (1972) 41.

    Google Scholar 

  39. R. Saigal, On a special linear complementarity problem, Opsearch 8 (1971) 201.

    Google Scholar 

  40. R. Saigal and R.E. Stone, Proper, reflecting and absorbing facets of complementary cones, Math. Progr. 31 (1985) 106.

    Article  Google Scholar 

  41. H. Samelson, R.M. Thrall and O. Wesler, A partition theorem for Euclideann-space, Proc. Amer. Math. Soc. 9 (1958) 805–807.

    Google Scholar 

  42. R.E. Stone, Linear complementarity problem with an invariant number of solutions, Math. Progr. 34 (1986) 265.

    Article  Google Scholar 

  43. M.J. Todd, Orientation in complementary pivot algorithms, Math. Oper. Res. 1 (1976) 54.

    Article  Google Scholar 

  44. L. Van der Heyden, A variable dimension algorithm for the linear complementarity problem, Math. Progr. 19 (1980) 328.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indian Statistical Institute, 110016, New Delhi, India

    S. R. Mohan

Authors
  1. S. R. Mohan
    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

Mohan, S.R. Degeneracy in linear complementarity problems: a survey. Ann Oper Res 46, 179–194 (1993). https://doi.org/10.1007/BF02096262

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096262

Keywords

Search

Navigation