Skip to main content
SpringerLink
Log in

Exceptional Family of Elements for a Variational Inequality Problem and its Applications

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P * -mappings is also obtained.

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. Eaves, B.C. (1971), On the basic theorem of complementarity problem, Math. Programming 1, 68-75.

    Google Scholar 

  2. Eaves, B.C. (1971), The linear complementarity problem, Management Science 17, 612-634.

    Google Scholar 

  3. Harker, P.T. and Pang, J.S. (1990), Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming 48, 161-220.

    Google Scholar 

  4. Harker, P.T. (1986), A note on the existence of traffic equilibria, Appl. Math. Computation 18, 277-283.

    Google Scholar 

  5. Harker, P.T. and Pang, J.S. (1994), Modelling and Computation of Equilibrium: A Variational Inequality Approach, Academic Press, New York.

    Google Scholar 

  6. Hartman, P. and Stampacchia, G. (1966), On some nonlinear elliptic differentiable functional equation, ACTA Math. 115, 271-310.

    Google Scholar 

  7. Isac, G., Bulavaski, V. and Kalashnikov, V. (1997), Exceptional families, topological degree and complementarity problems, J. Global Optim. 10, 207-225.

    Google Scholar 

  8. Karamardian, S. (1971), Generalized complementarity problem, J. Optim. Theory Appl. 8, 161-167.

    Google Scholar 

  9. Karamardian, S. (1976), An existence theorem for the complementarity problem, J. Optim. Theory Appl. 19, 227-232.

    Google Scholar 

  10. Karamardian, S. (1972), The complementarity problem, Math. Programming 2, 107-129.

    Google Scholar 

  11. Karamardian, S. (1976), Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18, 445-454.

    Google Scholar 

  12. Karamardian, S. and Schaible, S. (1990), Seven kinds of monotone maps, J. Optim. Theory Appl. 66, 37-46.

    Google Scholar 

  13. Kojima, M. (1975), A unification of the existence theorems of the nonlinear complementarity problem, Math. Programming 9, 257-277.

    Google Scholar 

  14. Kojima, M., Megiddo, N., Noma, T. and Yoshise, A. (1991), A Unified Approach to Interiorpoint Algorithm for Linear Complementarity Problems, Lecture Notes in Computer Science, Springer Verlag, Nr. 538.

  15. Lloyd, N.Q. (1978), Degree Theory, Cambridge University Press, Cambridge, England.

    Google Scholar 

  16. Megiddo, N. (1977), A monotone complementarity problem with feasible solutions but no complementarity solutions, Math. Programming 12, 131-132.

    Google Scholar 

  17. Miao, J.M. (1995), A quadratically convergent \(O((\kappa + 1) \sqrt n L)\)-iteration algorithm for the P-matrix linear complementarity problem, Math. Programming 69, 355-368.

    Google Scholar 

  18. Moré, J.J. (1974), Classes of functions and feasibility conditions in nonlinear complementarity problems, Math. Programming 6, 327-338.

    Google Scholar 

  19. Moré, J.J. (1974), Coercivity conditions in nonlinear complementarity problems, SIAM Rev. 16, 1-16.

    Google Scholar 

  20. Ortega, J.M. and Rheinholdt, W.C. (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.

    Google Scholar 

  21. Pang, J.S. and Yao, J.C. (1995), On a generalization of a normal map and equation, SIAM J. Control Optim. 33 168-184.

    Google Scholar 

  22. Pardalos, P.M., Ye, Y., Han, C.-G. and Kalinski, J. (1993), Solution of P-matrix linear complementarity problems using a potential reduction algorithm, SIAM J. Matrix Anal. & Appl. 14, 1048-1060.

    Google Scholar 

  23. Potra, F.A. and Sheng, R. (1997), A large-step infeasible-interior-point method for the P-matrix LCP, SIAM, J. Optim. 7, 318-335.

    Google Scholar 

  24. Potra, F.A. and Sheng, R. (1996), Predictor-corrector algorithm for solving P ./-matrix LCP from arbitrary positive points, Math. Programming 76, 223-244.

    Google Scholar 

  25. Sheng, R. and Potra, F.A. (1997), A quadratically convergent infeasible-interior-point algorithm for LCP with polynomial complexity, SIAM J. Optim. 7, 304-317.

    Google Scholar 

  26. Smith, T.E. (1984), A solution condition for complementarity problems, with an application to spatial price equilibrium, Appl. Math. Computation 15, 61-69.

    Google Scholar 

  27. Ye, Y. and Pardalos, P.M. (1991), A class of linear complementarity problems solvable in polynomial time, Linear Algebra and Its Applications 152, 3-17.

    Google Scholar 

  28. Zhao, Y.B. (1997), Exceptional families and finite-dimensional variational inequalities over polyhedral convex sets, Appl. Math. Computation 87, 111-126.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chinese Academy of Sciences, Institute of Applied Mathematics, P.O. Box 2734, Beijing, 100080, People's Republic of China

    Y.B. Zhao & J. Han

Authors
  1. Y.B. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Han
    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

Zhao, Y., Han, J. Exceptional Family of Elements for a Variational Inequality Problem and its Applications. Journal of Global Optimization 14, 313–330 (1999). https://doi.org/10.1023/A:1008202323884

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008202323884

Search

Navigation