Generalized linear complementarity problems treated without fixed-point theory

  • J. M. Borwein
Contributed Papers


We study the (monotone) linear complementarity problem in reflexive Banach space. The problem is treated as a quadratic program and shown to satisfy appropriate constraint qualifications. This leads to a theory of the generalized monotone linear complementarity problem which is independent of Brouwer's fixed-point theorem. Certain related results on linear systems are given. The final section concerns copositive operators.

Key Words

Complementarity dual inequality systems convex duality quadratic programming 


  1. 1.
    Karamardian, S.,Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Lemke, C. E.,A Brief Survey of Complementarity Theory, Constructive Approaches to Mathematical Models, Edited by C. V. Coffman and G. J. Fix, Academic Press, New York, New York, 1979.Google Scholar
  3. 3.
    Borwein, J. M.,Adjoint Process Duality, Mathematics of Operation Research Vol. 8, pp. 403–434, 1983.MATHCrossRefGoogle Scholar
  4. 4.
    Cottle, R. W.,Note on a Fundamental Theorem in Quadratic Programming, SIAM Journal of Applied Mathematics, Vol. 12, pp. 663–665, 1964.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cottle, R. W., andPang, J. S.,On Solving Linear Complementarity Problems as Linear Programs, Mathematical Programming Study, Vol. 7, pp. 88–107, 1978.MATHMathSciNetGoogle Scholar
  6. 6.
    Cryer, C. W. andDempster, M. A. H.,Equivalence of Linear Complementarity Problems and Linear Programs in Vector Lattice Hilbert Spaces, SIAM Journal on Control and Optimization, Vol. 18, pp. 76–90, 1980.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.MATHGoogle Scholar
  8. 8.
    Karamardian, S.,The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Holmes, R. B.,Geometric Functional Analysis, Springer-Verlag, New York, New York, 1975.MATHGoogle Scholar
  10. 10.
    More, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Eaves, B. C.,The Linear Complementarity Problem, Management Science, Vol. 17, pp. 612–634, 1971.MATHMathSciNetGoogle Scholar
  12. 12.
    Lemke, C. E.,Recent Results in Complementarity Problems, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, New York, New York, 1970.Google Scholar
  13. 13.
    Riddell, R. C.,Equivalence of Nonlinear Complementarity Problems and Least Element Problems in Banach Lattices, Mathematics of Operation Research, Vol. 6, pp. 462–474, 1981.MATHMathSciNetGoogle Scholar
  14. 14.
    Cottle, R. W., andDanzig, G. B.,Complementary Pivot Theory of Mathematical Programming, Linear Algebra and Its Applications, Vol. 1, pp. 103–175, 1968.MATHCrossRefGoogle Scholar
  15. 15.
    Browder, F.,Nonlinear Monotone Operators and Convex Sets in Banach Space, Bulletin of the American Mathematical Society, Vol. 71, pp. 780–785, 1965.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gwinner, J.,On Fixed Points and Variational Inequalities—A Circular Tour, Journal of Nonlinear Analysis: Theory, Methods, and Applications, Vol. 5, pp. 565–583, 1981.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    McLinden, L.,An Analogue of Moreau's Proximation Theorem, with Application to the Nonlinear Complementarity Problem, Pacific Journal of Mathematics, Vol. 88, pp. 101–161, 1980.MATHMathSciNetGoogle Scholar
  18. 18.
    McLinden, L.,The Complementarity Problem for Maximal Monotone Multifunctions, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, Chichester, England, 1980.Google Scholar
  19. 19.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.MATHGoogle Scholar
  20. 20.
    Borwein, J. M.,Convex Relations in Analysis and Optimization, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, pp. 335–377, 1981.Google Scholar
  21. 21.
    Frank, N., andWolfe, P.,An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly, Vol. 3, pp. 95–110, 1956.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Borwein, J. M.,A Lagrange Multiplier Theorem and a Sandwich Theorem for Convex Relations, Mathematica Scandia, Vol. 48, pp. 189–204, 1981.MATHGoogle Scholar
  23. 23.
    Robinson, S.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operation Research, Vol. 1, pp. 130–143, 1976.MATHGoogle Scholar
  24. 24.
    Mangasarian, O. L.,Characterizations of Bounded Solutions of Linear Complementarity Problems, Mathematical Programming Studies (to appear).Google Scholar
  25. 25.
    Bazaraa, M. S., andShetty, C. M.,Nonlinear Programming, Wiley, New York, New York, 1979.MATHGoogle Scholar
  26. 26.
    Aganagic, C., andCottle, R. W.,A Note on Q-Matrices, Mathematical Programming, Vol. 16, pp. 374–377, 1979.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Garcia, G. B.,Some Classes of Matrices in Linear Complementarity Theory, Mathematical Programming, Vol. 5, pp. 299–310, 1973.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • J. M. Borwein
    • 1
  1. 1.Mathematics DepartmentDalhousie UniversityHalifax

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