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A fixed-point representation of the generalized complementarity problem

  • S. C. Fang
  • E. L. Peterson
Contributed Papers
  • 58 Downloads

Abstract

Eaves and Kojima have separately provided fixed-point representations of the standard complementarity problem. Although the mappings used to describe their representations appear to be different, this paper shows they are essentially the same, a unification that is accomplished via a geometric programming argument in the context of a more general complementarity problem.

Key Words

Complementarity fixed points dual cones geometric programming 

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References

  1. 1.
    Eaves, B. C.,On the Basic Theory of Complementarity, Mathematical Programming, Vol. 1, pp. 68–75, 1971.Google Scholar
  2. 2.
    Browder, F. E.,On Continuity of Fixed Points under Deformation of Continuous Mappings, Summa Brasiliensis Mathematicae, Vol. 4, pp. 183–191, 1960.Google Scholar
  3. 3.
    Kojima, N.,A Unification of the Existence Theorems of the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 9, pp. 257–277, 1975.Google Scholar
  4. 4.
    Megiddo, N., andKojima, M.,On the Existence and Uniqueness of Solutions in Nonlinear Complementarity Theory, Mathematical Programming, Vol. 12, pp. 110–130, 1977.Google Scholar
  5. 5.
    Peterson, E. L.,Optimality Conditions in Generalized Geometric Programming, Journal of Optimization Theory and Applications, Vol. 26, pp. 3–13, 1978.Google Scholar
  6. 6.
    Peterson, E. L.,Geometric Programming, SIAM Review, Vol. 18, pp. 1–51, 1976.Google Scholar
  7. 7.
    Hall, M., andPeterson, E. L.,Traffic Equilibria Analyzed via Geometric Programming, Proceedings of the International Symposium on Traffic Equilibrium Methods, Edited by M. Florian, Springer-Verlag, Berlin, Germany, pp. 53–109, 1976.Google Scholar
  8. 8.
    Peterson, E. L.,The Conical Duality and Complementarity of Price and Quantity for Multicommodity Spatial and Temporal Network Allocation Probloems, Northwestern University, Center for Mathematical Studies in Economics and Management Science, Discussion Paper No. 207, 1976.Google Scholar
  9. 9.
    Saigal, R.,Extension of the Generalized Complementarity Problem, Mathematics of Operations Research, Vol. 1, pp. 260–266, 1976.Google Scholar
  10. 10.
    Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.Google Scholar
  11. 11.
    Karamardian, S.,Complementarity Problems over Cones with Monotone and Pseudomotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.Google Scholar
  12. 12.
    Habetler, G. J., andPrice, A. L.,Existence Theory for Generalized Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 7, pp. 223–239, 1971.Google Scholar
  13. 13.
    Lemke, C. E.,Bimatrix Equilibrium Points and Mathematical Programming, Management Science, Vol. 11, pp. 681–689, 1965.Google Scholar
  14. 14.
    Cottle, R. W.,Nonlinear Programs with Positively Bounded Jacobians, SIAM Journal on Applied Mathematics, Vol. 14, pp. 147–158, 1966.Google Scholar
  15. 15.
    Cottle, R. W., andDantzig, G. B.,Complementary Pivot Theory of Mathematical Programming, Linear Algebra and Its Applications, Vol. 1, pp. 103–125, 1968.Google Scholar
  16. 16.
    Eaves, B. C.,The Linear Complementarity Problem, Management Science, Vol. 17, pp. 612–634, 1971.Google Scholar
  17. 17.
    Merrill, O. H.,Applications and Extensions of an Algorithm That Computes Fixed Points of Certain Upper Semicontinuous Point-to-Set Mappings, University of Michigan, PhD Thesis, 1972.Google Scholar
  18. 18.
    Saigal, R.,On the Class of Complementarity Cones and Lemke's Algorithm, SIAM Journal on Applied Mathematics, Vol. 23, pp. 46–60, 1972.Google Scholar
  19. 19.
    Eaves, B. C., andSaigal, R.,Homotopies for Computing Fixed Points in Unbounded Regions, Mathematical Programming, Vol. 3, pp. 225–237, 1972.Google Scholar
  20. 20.
    Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.Google Scholar
  21. 21.
    Saigal, R., andSimon, C. P.,Generic Properties of the Complementarity Problem, Mathematical Programming, Vol. 4, pp. 324–335, 1973.Google Scholar
  22. 22.
    More, J. J.,Classes of Functions and Feasibility Conditions in Nonlinear Complementarity Problems, Mathematical Programming, Vol. 6, pp. 327–338, 1974.Google Scholar
  23. 23.
    More, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.Google Scholar
  24. 24.
    Fisher, M., andGould, F. J.,A Simplicial Algorithm for the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 6, pp. 281–300, 1974.Google Scholar
  25. 25.
    Kojima, M.,Computing Methods for Solving the Complementarity Problem, Keio Engineering Reports, Vol. 27, pp. 1–41, 1974.Google Scholar
  26. 26.
    Moreau, J. J.,Decomposition Orthogonale d'un Espace Hilbertien selon Deux Cones Mutuellement Polaries, Comptes Rendus, Vol. 255, pp. 238–240, 1962.Google Scholar

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • S. C. Fang
    • 1
  • E. L. Peterson
    • 2
  1. 1.AT&T Technologies, Engineering Research CenterPrinceton
  2. 2.Department of Mathematics and Graduate Program in Operations ResearchNorth Carolina State UniversityRaleigh

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