On quasidifferential kernels

Abstract

ZUN-QUAN XIA ON QUASIDIFFERENTIAL KERNELSThe purpose of this paper is to explore the structure of guasidifferential kernels of a class of quasidifferentiable functions in the sense of Demyanov and Rubinov. The corresponding properties and operations concerning kernels are given.

Representative of Quasidifferentials and Its Formula for a Quasidifferentiable Function

“…As we know, the uppersemicontinuity of the quasidifferential is required in many existing algorithms of quasidifferentiable optimization [3] and of quasidifferentiable equations [16]. Many publications dealt with this problem, see for instance [8,9,11,14,15,17].…”

“…For the same purpose, Xia [14,15] introduced the notion of the kernelled quasidifferential. The following pair of sets…”

Research on Adjoint Kernelled Quasidifferential

“…The problem was for the first time considered in a discussion at IIASA, by Demyanov and Xia in 1984 [4]. There were many reports and publications mentioning or dealing with this subject from different points of view (see, for instance, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], etc.). Pallaschke et al [18] introduced the notion of the minimal quasidifferential and proved the existence of equivalent minimal quasidifferential.…”

“…For the same purpose, Xia [24,25] introduced the notion of the kernelled quasidifferential. It was proved that = ⋂…”

“…Although the kernelled quasidifferential is known to have good algebraic properties and geometric structure (see [25]), it is still not very convenient for calculating the kernelled quasidifferentials of − and min{ | ∈ a finite index set }, where and are kernelled quasidifferentiable functions. Hence, in this paper, the notion of adjoint kernelled quasidifferential, which is well-defined for − and min{ | ∈ }, is employed as a representative of the equivalence class of quasidifferentials.…”