Star-Kernels and Star-Differentials in Quasidifferential Analysis

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.