Skip to main content
SpringerLink
Log in

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C 1 and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C 2 and C 1 nonlinear complementarity problems.

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. GOWDA, M. S., and RAVINDRAN, G., Algebraic Univalence Theorems for Nonsmooth Functions, Journal of Mathematical Analysis and Applications, Vol. 252, pp. 917-935, 2000.

    Google Scholar 

  2. SONG, Y., GOWDA, M. S., and RAVINDRAN, G., On Characterizations of P and P 0 Properties in Nonsmooth Functions, Mathematics of Operations Research, Vol.25, pp. 400-408, 2000.

    Google Scholar 

  3. TAWHID, M. A., and GOWDA, M. S., On Two Applications of H-Differentiability to Optimization and Complementarity Problems, Computational Optimization and Applications, Vol. 17, pp. 279-299, 2000.

    Google Scholar 

  4. GOWDA, M. S., A Note on H-Differentiable Functions, Department of Mathematics and Statistics, University of Maryland at Baltimore County, Baltimore, Maryland, 1998.

    Google Scholar 

  5. JEYAKUMAR, V., and LUC, D. T., Approximate Jacobians Matrices for Nonsmooth Continuous Maps and C1-Optimization, SIAM Journal on Control and Optimization, Vol. 36, pp. 1815-1832, 1998.

    Google Scholar 

  6. JIANG, H., and QI, L., Local Uniqueness and Convergence of Iterative Methods for Nonsmooth Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 196, pp. 314-331, 1995.

    Google Scholar 

  7. PANG, J. S., Complementarity Problems, Handbook on Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Boston, Massachusetts, 1994.

    Google Scholar 

  8. PENG, J., KANZOW, C., and FUKUSHIMA, M., A Hybrid Newton Method for Solving Box-Constrained Variational Inequality Problems via the D-Gap Function, Optimization Methods and Software, Vol. 10, pp. 687-710, 1999.

    Google Scholar 

  9. KYPARISIS, J., Uniqueness and Differentiability of Solutions of Parametric Nonlinear Complementarity Problems, Mathematical Programming, Vol. 36, pp. 105-113, 1986.

    Google Scholar 

  10. MANGASARIAN, O. L., Locally Unique Solutions of Quadratic Programs, Linear and Nonlinear Complementarity Problems, Mathematical Programming, Vol. 19, pp. 200-212, 1980.

    Google Scholar 

  11. MORÉ, J. J., and RHEINBOLDT, W. C., On P and S Functions and Related Classes of N-Dimensional Nonlinear Mappings, Linear Algebra and Its Applications, Vol. 6, pp. 45-68, 1973.

    Google Scholar 

  12. ROCKAFELLAR, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  13. ROCKAFELLAR, R. T., and WETS, R. J. B., Variational Analysis, Grundlehren der Mathematischen Wissenschaften, Springer Verlag, Berlin, Germany, 1998.

    Google Scholar 

  14. CLARKE, F. H., Optimization and Nonsmooth Analysis, Wiley, New York, NY, 1983; Reprinted by SIAM, Philadelphia, Pennsylvania, 1990.

    Google Scholar 

  15. MIFFLIN, R., Semismooth and Semiconvex Functions in Constrained Optimization, SIAM Journal on Control and Optimization, Vol. 15, pp. 952-972, 1977.

    Google Scholar 

  16. QI, L., Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operations Research, Vol. 18, pp. 227-244, 1993.

    Google Scholar 

  17. QI, L., and SUN, J., A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353-367, 1993.

    Google Scholar 

  18. QI, L., C-Differentiability, C-Differential Operators, and Generalized Newton Methods, Research Report, School of Mathematics, University of New South Wales, Sydney, New South Wales, Australia, 1996.

    Google Scholar 

  19. ROOS, C., and TERLAKY, T., Nonlinear Optimization, Faculty of Information Technology and Systems, Delft University of Technology, Delft, Holland, 1998.

    Google Scholar 

  20. STOER, J., and WITZGALL, C., Convexity and Optimization in Finite Dimensions, Vol. 1, Springer Verlag, Berlin, Germany, 1970.

    Google Scholar 

  21. HARKER, P. T., and PANG, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161-220, 1990.

    Google Scholar 

  22. TOBIN, R. L. Sensitivity Analysis for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 48, pp. 191-204, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt

    M.A. Tawhid (Assistant Professor)

Authors
  1. M.A. Tawhid
    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

Tawhid, M. On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability. Journal of Optimization Theory and Applications 113, 149–164 (2002). https://doi.org/10.1023/A:1014813415372

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1014813415372

Search

Navigation