(h，φ)-凸函数与(h，φ)-Lipschitz函数的一些广义微分性质

Properties for Generalized Differential of (h,φ)-convex Functions and (h,φ)-Lipschitz Function

利用函数f与它的对应函数f(t)=φ(f(h^(-1)(t)))之间的关系,研究了(h,φ)-凸函数和(h,φ)- Lipschitz函数的广义方向导数,得到了R^n上连续(h,φ)-凸函效的广义方向导数的有限性、上半连续性以及估值不等式.在f是R^n上的(h,φ)-凸函数的假设下,给出了f为局部(h,φ)-Lipschitz的一个充分必要条件.并讨论了R^n上的(h,φ)-凸函数和(h,φ)-Lipschitz函数的关系,得到了(h,φ)-凸函数的广义次微分的几个基本性质.

By making use of the relationship between a function and f its corresponding function f（ t ） = φ（f（h^-1（t））, this paper studied some properties for generalized directional derivatives of （h,φ）-convex functions and （h, φ）-Lipschitz functions. It is shown that generalized directional derivative of a continuous （h, φ）- convex function defined on Rn is finite, upper semicontinuous and satisfies an inequality. A necessary and sufficient condition characterizing （h,φ）-Lipschitz functions f defined on Rn is obtained under the assumption that f is （h, φ）- convex. As applications, the relation between （h, φ）-convex functions and （h, φ）-Lipschitz functions, and some fundamental properties of the generalized subdifferential of （h,φ）- convex functions are presented.