Complementarity Pivot Methods

  • G. Isac
  • V. A. Bulavsky
  • V. V. Kalashnikov
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 63)

Abstract

This chapter deals with pivot methods that constituted the first and important instrument in solving Linear Complementarity Problems. Developed under the impact of the simplex method, they revealed many crucial differences between the linear programming and complementarity problems.

Keywords

Complementarity Problem Basic Solution Linear Complementarity Problem Direct Algorithm Complementary Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Chandrasekaran R. A special case of the complementarity pivot problem. Opsearch 1970; 7: 263–268.MathSciNetGoogle Scholar
  2. Cottle RW. 1. Nonlinear programs with positively bounded Jacobians. SIAM J. Appl. Math. 1966; 14:147–158.MathSciNetMATHCrossRefGoogle Scholar
  3. Cottle RW. 2. On a problem in linear inequalities. J. London Math. Soc. 1968; 43:378–384.MathSciNetMATHCrossRefGoogle Scholar
  4. Cottle RW, and Dantzig GB. Complementarity pivot theory of mathematical programming. Linear Algebra and Its Applications. 1968; 1:103–125.MathSciNetMATHCrossRefGoogle Scholar
  5. Eaves, B.C. 1. “The linear complementarity problem in mathematical programming,” Research Report 69–4, Operations Research House, Stanford University, Stanford, California, 1969.Google Scholar
  6. Eaves, B.C. 2. The linear complementarity problem. Management Science. 1971; 17:612–634.MathSciNetMATHCrossRefGoogle Scholar
  7. Eaves, B.C. 3. On quadratic programming. Management Science. 1971; 17: 698–711.MATHCrossRefGoogle Scholar
  8. Eaves, B.C. 4. ‘A short course in solving equations with PL homotopies.’- In: Nonlinear Programming, Vol. IX, SIAM-AMS Proceedings, R.W. Cottle and C.E. Lemke, eds. American Mathematical Society, Providence, 1976.Google Scholar
  9. Fiedler M, and Ptak V. On matrices with non-positive off-diagonal elements and positive principal minors. Czechoslovak Mathematical Journal. 1962; 12:382–400.MathSciNetCrossRefGoogle Scholar
  10. Garcia CB. Some classes of matrices in linear complementarity theory. Math. Programming. 1973; 5:299–310.MathSciNetMATHCrossRefGoogle Scholar
  11. Habetler CJ, and Kostreva MM. On a direct algorithm for nonlinear complementarity problems. SIAM J. Control Optim. 1978; 16: 504–511.MathSciNetMATHCrossRefGoogle Scholar
  12. Habetler CJ, and Price AL. Existence theory for generalized nonlinear conplementarity problems. J. Optim. Theory Appl. 1971; 8:161–168.MathSciNetCrossRefGoogle Scholar
  13. Herstein, I.N., Topics in Algebra. New York: Blaisdell, 1964.MATHGoogle Scholar
  14. Kostreva, M.M. 1. “Direct algorithms for complementarity problems,” Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, N.Y., 1976.Google Scholar
  15. Kostreva, M.M. 2. Block pivot methods for solving the complementarity problem. Linear Algebra and its Applications. 1978; 21: 207–215.MathSciNetMATHCrossRefGoogle Scholar
  16. Moré J. Coercivity conditions in nonlinear complementarity problems. SIAM Review. 1974; 16:1–15.MathSciNetMATHCrossRefGoogle Scholar
  17. Moré J, and Rheinboldt W. On P- and S-functions and related classes of n-dimensional nonlinear mappings. Linear Algebra and Applications. 1973; 6:45–68.MATHCrossRefGoogle Scholar
  18. Ingleton AW. A problem in linear inequalities. Proc. London Math. Soc, Third Series. 1966; 16: 519–536.MathSciNetMATHCrossRefGoogle Scholar
  19. Lemke CB. 1. Bimatrix equilibrium points and mathematical programming. Management Science. 1965; 7:681–689.MathSciNetCrossRefGoogle Scholar
  20. Lemke CB. 2. “On complementarity pivot theory,” RPI Math Report No. 76, Rensselaer Polytechnic Institute, Troy, New York, 1967.Google Scholar
  21. Lemke CB. 3. ‘On Complementarity Pivot Theory’.- In: Mathematics of the Decisidn Sciences, G.B. Dantzig and A.F. Veinott Jr., eds., American Mathematical Society, Providence, Rhode Island, 1968.Google Scholar
  22. Murty, K.G. 1. “On the number of solutions to the complementarity quadratic programming problem,” Doctoral Dissertation, Engineering Science, University of California, Berkeley, 1968.Google Scholar
  23. Murty, K.G. 2. Linear Complementarity, Linear and Nonlinear Programming. Berlin: Heldermann Verlag, 1988.MATHGoogle Scholar
  24. Nash J. Non-cooperative games. Annals of Mathematics. 1951; 54:286–295.MathSciNetMATHCrossRefGoogle Scholar
  25. Ortega, J.M, and Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press, 1970.MATHGoogle Scholar
  26. Tamir A. Minimality and complementarity properties associated with Z-functioris and M-functions. Math. Programming. 1974; 7:17–31.MathSciNetMATHCrossRefGoogle Scholar
  27. Todd, M.J. 1. “Dual families of linear programs,” Technical Report No. 197, Department of Operations Research, Cornell University, 1973.Google Scholar
  28. Todd, M.J. 2. Extensions of Lemke’s algorithm for the linear complementarity problem. J. Optim. Theory Appl. 1976; 20:397–416.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • G. Isac
    • 1
  • V. A. Bulavsky
    • 2
  • V. V. Kalashnikov
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Central Economics Institute (CEMI) of Russian Academy of SciencesMoscowRussia

Personalised recommendations