Skip to main content
SpringerLink
Log in

Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions

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

Abstract

New concepts of semistrict quasimonotonicity and strict quasimonotonicity for multivalued maps are introduced. It is shown that a locally Lipschitz map is (semi)strictly quasiconvex if and only if its Clarke subdifferential is (semi)strictly quasimonotone. Finally, an existence result for the corresponding variational inequality problem is 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. Mangasarian, O., Nonlinear Programming, New York, McGraw-Hill, New York, New York, 1969.

    Google Scholar 

  2. Daniilidis, A., Hadjisavvas, N., and Schaible, S., Connectedness of the Efficient Set for Three-Objective Quasiconcave Maximization Problems, Journal of Optimization Theory and Applications, Vol. 93, pp. 517–524, 1997.

    Google Scholar 

  3. Benoist, J., Connectedness of the Efficient Set for Strictly Quasiconcave Sets, Journal of Optimization Theory and Applications, Vol. 96, pp. 627–654, 1998.

    Google Scholar 

  4. Karamardian, S., Complementatity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.

    Google Scholar 

  5. Karamardian, S., and Schaible, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.

    Google Scholar 

  6. Luc, D. T., Characterizations of Quasiconvex Functions, Bulletin of the Australian Mathematical Society, Vol. 48, pp. 393–406, 1993.

    Google Scholar 

  7. Aussel, D., Corvellec, J. N., and Lassonde, M., Subdifferential Characterization of Quasiconvexity and Convexity, Journal of Convex Analysis, Vol, 1, pp. 195–201, 1994.

    Google Scholar 

  8. Penot, J. P., Generalized Convexity in the Light of Nonsmooth Analysis, Recent Developments in Optimization, Edited by R. Durier and C. Michelot, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 429, pp. 269–290, 1995.

    Google Scholar 

  9. Penot, J. P., and Quang, P. H., Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps, Journal of Optimization Theory and Applications, Vol. 92, pp. 343–356, 1997.

    Google Scholar 

  10. Hadjisavvas, N., and Schaible, S., On Strong Pseudomonotonicity and (Semi) strict Quasimonotonicity, Journal of Optimization Theory and Applications, Vol. 79, pp. 139–155, 1993.

    Google Scholar 

  11. Hadjisavvas, N., and Schaible, S., On Strong Pseudomonotonicity and (Semi) strict Quasimonotonicity, Errata Corrige, Journal of Optimization Theory and Applications, Vol. 85, pp. 741–742, 1995.

    Google Scholar 

  12. Konnov, I., On Quasimonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 99, pp. 165–181, 1998.

    Google Scholar 

  13. Rockafellar, R. T., Generalized Directional Derivetives and Subgradients of Nonconvex Functions, Canadian Journal of Mathematics, Vol. 32, pp. 257–280, 1980.

    Google Scholar 

  14. Avriel, M., Diewert, W. E., Schaible, S., and Zang, I., Generalized Concavity, Plenum Publishing Corporation, New York, New York, 1988.

    Google Scholar 

  15. Aussel, D., Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: A Unified Approach, Journal of Optimization Theory and Applications, Vol. 97, pp. 29–45, 1998.

    Google Scholar 

  16. Zagrodny, D., Appropriate Mean-Value Theorem for Upper Subderivatives, Nonlinear Analysis, Vol. 12, pp. 1413–1428, 1988.

    Google Scholar 

  17. Minty, G., Monotone (Nonlinear) Operators in Hilbert Space, Duke Mathematical Journal, Vol. 29, pp. 341–346, 1962.

    Google Scholar 

  18. Yao, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994.

    Google Scholar 

  19. Hadjisavvas, N., and Schaible, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 95–111, 1996.

    Google Scholar 

  20. Daniilidis, A., and Hadjisavvas, N., On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity, Journal of Mathematical Analysis and Applications (to appear).

  21. Fan, K., A Generalization of Tychonoff's Fixed-Point Theorem, Mathematische Annlen, Vol. 142, pp. 305–310, 1961.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of the Aegean, Samos, Greece

    A. Daniilidis (Researcher)

  2. Department of Mathematics, University of the Aegean, Samos, Greece

    N. Hadjisavvas (Professor)

Authors
  1. A. Daniilidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Hadjisavvas
    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

Daniilidis, A., Hadjisavvas, N. Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions. Journal of Optimization Theory and Applications 102, 525–536 (1999). https://doi.org/10.1023/A:1022693822102

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022693822102

Search

Navigation