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Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps

Author

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  • J.-P. Penot

    (Université de Pau)

  • P. H. Quang

    (Institute of Mathematics)

Abstract

We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps is introduced; it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:92:y:1997:i:2:d:10.1023_a:1022659230603
    DOI: 10.1023/A:1022659230603
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    1. Correa Romar, 2014. "Mathematical Foci," Mathematical Economics Letters, De Gruyter, vol. 2(1-2), pages 1-7, August.
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    Cited by:

    1. N. Hadjisavvas & S. Schaible & N.-C. Wong, 2012. "Pseudomonotone Operators: A Survey of the Theory and Its Applications," Journal of Optimization Theory and Applications, Springer, vol. 152(1), pages 1-20, January.
    2. Vsevolod I. Ivanov, 2020. "Characterization of Radially Lower Semicontinuous Pseudoconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 368-383, February.
    3. 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.
    4. Hoa T. Bui & Pham Duy Khanh & Thi Tu Trinh Tran, 2019. "Characterizations of Nonsmooth Robustly Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 775-786, March.
    5. Ghirardato, Paolo & Siniscalchi, Marciano, 2018. "Risk sharing in the small and in the large," Journal of Economic Theory, Elsevier, vol. 175(C), pages 730-765.
    6. M. Castellani & M. Giuli, 2010. "On Equivalent Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 157-168, October.
    7. 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.
    8. Xin He & Nan-jing Huang & Xue-song Li, 2022. "Modified Projection Methods for Solving Multi-valued Variational Inequality without Monotonicity," Networks and Spatial Economics, Springer, vol. 22(2), pages 361-377, June.
    9. D. Aussel, 1998. "Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 29-45, April.

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