Geodesic
Star-Kernels and Star-Differentials in Quasidifferential Analysis
Journal of convex analysis, Tome 9 (2002) no. 1, pp. 139-158.

Voir la notice de l'article provenant de la source Heldermann Verlag

This paper is devoted to the study of quasidifferential structure. Three concepts, kernelled quasidifferential, star-kernel and star-differential, are proposed. The kernelled quasidifferential is used to describe a special class of quasidifferentiable functions, which covers convex and concave functions. A sufficiency theorem and a sufficiency and necessity theorem for a quasi-kernel being a kernelled quasidifferential are proved. The notion of star-kernel is employed if the quasi-kernel is not a kernelled quasidifferntial. The existence theorem for a star-kernel of a quasidifferentiable function is established, which shows that the star-kernel is a pair of star-shaped sets and the sub-/super-derivative is expressed by the gauge of a star-shaped set. The notion of star-differential is used to describe the differential of the class of directionally differentiable functions which contains the class of quasidifferentiable functions. A star-differential is also a pair of star-shaped sets and its operational properties are favourable. A representative of the star-differential can be easily obtained by decomposing the directional derivative into the difference of its positive and negative parts.
Classification : 90C30
Mots-clés : Quasidifferentiable function, directional derivative, kernelled quasidifferential, star-differential, star-kernel, star-shaped set 
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L.-W. Zhang; Z.-Q. Xia; Y. Gao; M.-Z. Wang. Star-Kernels and Star-Differentials in Quasidifferential Analysis. Journal of convex analysis, Tome 9 (2002) no. 1, pp. 139-158. http://geodesic.mathdoc.fr/item/JCA_2002_9_1_JCA_2002_9_1_a6/