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Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

Author

Listed:
  • A. Daniilidis

    (Universitat Autònoma de Barcelona)

  • Y. Garcia Ramos

    (Université des Antilles et de la Guyane
    Instituto de Matematica y Ciencias Afines)

Abstract

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T : X⇉X * relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.

Suggested Citation

  • A. Daniilidis & Y. Garcia Ramos, 2007. "Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 37-48, April.
    • Handle: RePEc:spr:joptap:v:133:y:2007:i:1:d:10.1007_s10957-007-9182-4
    DOI: 10.1007/s10957-007-9182-4
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    References listed on IDEAS

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    1. A. Daniilidis & N. Hadjisavvas, 1999. "Characterization of Nonsmooth Semistrictly Quasiconvex and Strictly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 525-536, September.
    2. J.-P. Penot & P. H. Quang, 1997. "Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 92(2), pages 343-356, February.
    3. J. Benoist & N. Popovici, 2001. "Contractibility of the Efficient Frontier of Three-Dimensional Simply-Shaded Sets," Journal of Optimization Theory and Applications, Springer, vol. 111(1), pages 81-116, October.
    4. Giorgio Giorgi & Sándor Komlósi, 1995. "Dini derivatives in optimization — Part III," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 18(1), pages 47-63, March.
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    Cited by:

    1. Vsevolod Ivanov, 2013. "Characterizations of pseudoconvex functions and semistrictly quasiconvex ones," Journal of Global Optimization, Springer, vol. 57(3), pages 677-693, November.

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