Journal of Optimization Theory and Applications

, Volume 71, Issue 3, pp 517–534 | Cite as

The generalized order complementarity problem

  • G. Isac
  • M. Kostreva
Contributed Papers

Abstract

Given an ordered Banach Space (E,K) andm functionsf1,f2,...,fm:EE, the generalized order complementarity problem associated with {fi} andK is to findx0K such thatfi(x0)∈K,i=1,...,m, and Λ (x0,f1(x0),...,fm(x0))=0. The problem is shown to be equivalent to several fixed-point problems and equivalent to the order complementarity problem studied by Borwein and Dempster and by Isac. Existence and uniqueness of solutions and least-element theory are shown in the spacesC(Ω, ℝ) andLp(Ω, μ). For general locally convex spaces, least-element theory is derived, existence is proved, and an algorithm for computing a solution is presented. Applications to the mixed lubrication theory of fluid mechanics are described.

Key Words

Complementarity ordered spaces fixed points lubrication theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bershchanskii, Ya. M., andMeerov, M. V.,The Complementarity Problem: Theory and Methods of Solution, Automation and Remote Control, Vol. 44, pp. 687–510, 1983.Google Scholar
  2. 2.
    Cottle, R. W.,Complementarity and Variational Problems, Symposia Mathematica, Vol. 19, pp. 177–208, 1976.Google Scholar
  3. 3.
    Isac, G.,Problèmes de Complémentarité (en Dimension Infinie), Department of Mathematics, Université de Limoges, Limoges, France, 1985.Google Scholar
  4. 4.
    Borwein, J. M., andDempster, M. A. H.,The Linear Order Complementarity Problem, Mathematics of Operations Research, Vol. 14, pp. 534–558, 1989.Google Scholar
  5. 5.
    Isac, G.,Complementarity Problem and Coincidence Equations on Convex Cones, Bollettino dell'Unione Matematica Italiana, Series B, Vol. 6, pp. 925–943, 1986.Google Scholar
  6. 6.
    Isac, G.,Fixed-Point Theory, Coincidence Equations on Convex Cones, and Complementarity Problem, Contemporary Mathematics, Vol. 72, pp. 139–155, 1988.Google Scholar
  7. 7.
    Kostreva, M. M.,Elasto-Hydrodynamic Lubrication: A Nonlinear Complementarity Problem, International Journal of Numerical Methods in Fluids, Vol. 4, pp. 377–397, 1984.Google Scholar
  8. 8.
    Kostreva, M. M.,Recent Results on Complementarity Models for Engineering and Economics, INFOR, Vol. 28, pp. 324–334, 1990.Google Scholar
  9. 9.
    Oh, K. P.,The Formulation of the Mixed Lubrication Problem as a Generalized Nonlinear Complementarity Problem, Transactions of the ASME, Journal of Tribology, Vol. 108, pp. 598–604, 1986.Google Scholar
  10. 10.
    Cottle, R. W., andDantzig, G. B.,A Generalization of the Linear Complementarity Problem, Journal of Combinatorial Theory, Vol. 8, pp. 79–90, 1970.Google Scholar
  11. 11.
    Mangasarian, O. L.,Generalized Linear Complementarity Problems as Linear Programs, Operations Research Verfahren, Vol. 31, pp. 393–402, 1979.Google Scholar
  12. 12.
    Luxemburg, W. A., andZaanen, A. C.,Riesz Spaces, Vol. 1, North-Holland Publishing Company, Amsterdam, The Netherlands, 1971.Google Scholar
  13. 13.
    Peressini, A. L.,Ordered Topological Vector Spaces, Harper and Row, New York, New York, 1967.Google Scholar
  14. 14.
    Fujimoto, T.,Nonlinear Complementarity Problems in a Function Space, SIAM Journal on Control and Optimization, Vol. 18, pp. 621–623, 1980.Google Scholar
  15. 15.
    Birkhoff, G.,Lattice Theory, American Mathematical Society, New York, New York, 1967.Google Scholar
  16. 16.
    Tarski, A.,A Lattice-Theoretical Fixed-Point Theorem and Its Applications, Pacific Journal of Mathematics, Vol. 5, pp. 285–309, 1955.Google Scholar
  17. 17.
    Isac, G., Un Théorème de Point Fixe, Application à la Comparaison des Equations Differentielles dans les Espaces de Banach Ordonnes, Libertas Mathematica, Vol. 1, pp. 75–89, 1981.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • G. Isac
    • 1
  • M. Kostreva
    • 2
  1. 1.Départment de MathématiquesCollège Militaire Royal de Saint-JeanCanada
  2. 2.Department of Mathematical SciencesClemson UniversityClemson

Personalised recommendations