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Variational Analysis of Paraconvex Multifunctions

Author

Listed:
  • Huynh Ngai

    (Quy Nhon University)

  • Nguyen Huu Tron

    (Quy Nhon University)

  • Nguyen Vu

    (Quy Nhon University)

  • Michel Théra

    (XLIM UMR-CNRS 7252, Université de Limoges
    Information Technology and Physical Sciences, Federation University)

Abstract

Our aim in this article is to study the class of so-called $$\rho -$$ ρ - paraconvex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function $$\rho :X\rightarrow [0,+\infty )$$ ρ : X → [ 0 , + ∞ ) satisfying some suitable conditions. This class of multifunctions generalizes the class of $$\gamma -$$ γ - paraconvex multifunctions with $$\gamma >1$$ γ > 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study.

Suggested Citation

  • Huynh Ngai & Nguyen Huu Tron & Nguyen Vu & Michel Théra, 2022. "Variational Analysis of Paraconvex Multifunctions," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 180-218, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-022-02021-2
    DOI: 10.1007/s10957-022-02021-2
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    References listed on IDEAS

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    1. Huynh Ngai & Jean-Paul Penot, 2007. "Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators," Lecture Notes in Economics and Mathematical Systems, in: Generalized Convexity and Related Topics, pages 379-397, Springer.
    2. Jean-Philippe Vial, 1983. "Strong and Weak Convexity of Sets and Functions," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 231-259, May.
    3. VIAL, Jean-Philippe, 1983. "Strong and weak convexity of sets and functions," LIDAM Reprints CORE 529, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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