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Journal of Convex Analysis 09 (2002), No. 1, 139--158
Copyright Heldermann Verlag 2002

Star-Kernels and Star-Differentials in Quasidifferential Analysis

Li-Wei Zhang
Institute of Computational Mathematics and Scientific / Engineering Computing, Chinese Academy of Sciences, P. O. Box 2719, 100080 Beijing, P. R. China

Zun-Quan Xia
CORA, Dept. of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China

Yan Gao
School of Management, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China

Ming-Zheng Wang
CORA, Dept. of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China

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.

Keywords: Quasidifferentiable function, directional derivative, kernelled quasidifferential, star-differential, star-kernel, star-shaped set.

MSC: 90C30

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