
Journal of Convex Analysis 09 (2002), No. 1, 139158 Copyright Heldermann Verlag 2002 StarKernels and StarDifferentials in Quasidifferential Analysis LiWei Zhang Institute of Computational Mathematics and Scientific / Engineering Computing, Chinese Academy of Sciences, P. O. Box 2719, 100080 Beijing, P. R. China zlw@lsec.cc.ac.cn ZunQuan 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 MingZheng 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, starkernel and stardifferential, 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 quasikernel being a kernelled quasidifferential are proved. The notion of starkernel is employed if the quasikernel is not a kernelled quasidifferntial. The existence theorem for a starkernel of a quasidifferentiable function is established, which shows that the starkernel is a pair of starshaped sets and the sub/superderivative is expressed by the gauge of a starshaped set. The notion of stardifferential is used to describe the differential of the class of directionally differentiable functions which contains the class of quasidifferentiable functions. A stardifferential is also a pair of starshaped sets and its operational properties are favourable. A representative of the stardifferential 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, stardifferential, starkernel, starshaped set. MSC: 90C30 [ Fulltextpdf (562 KB)] 