Abstract

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 quasidifferential. 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Li-Wei Zhang

Institute of Computational Mathematics and Scientific / Engineering Computing, Chinese Academy of Sciences, P. O. Box 2719, 100080 Beijing, P. R. China
and: CORA, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

zlw@lsec.cc.ac.cn

Zun-Quan Xia

CORA, Dept. of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China

zqxiazhh@dlut.edu.cn

Yan Gao

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

gaoyan1962@263.net

Ming-Zheng Wang

CORA, Dept. of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China

Keywords

  • Quasidifferentiable function
  • directional derivative
  • kernelled quasidifferential
  • star-differential
  • star-kernel
  • star-shaped set

Mathematics Subject Classification

90C30

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Suggested citation

L.-W. Zhang, Z.-Q. Xia, Y. Gao, M.-Z. Wang. “Star-Kernels and Star-Differentials in Quasidifferential Analysis.” Journal of Convex Analysis 9 (2002), No. 1, 139–158. https://doi.org/10.68381/jca09007

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.