Variational Analysis of Paraconvex Multifunctions

Huynh Ngai (), Nguyen Huu Tron (), Nguyen Vu () and Michel Théra ()
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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

Journal of Optimization Theory and Applications, 2022, vol. 193, issue 1, No 11, 180-218

Abstract: 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.

Keywords: Weak convexity; Lower $$C^2$$ C 2 functions; Paraconvexity; Paramonotonicity; Approximate convex function; Fréchet subdifferential; Fréchet normal cone; Coderivatives; Fuzzy mean value theorem; 49J52; 49J53; 90C30 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-022-02021-2

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