(<i>h，&#966;</i>)-凸函数与（<i>h，&#966;</i>）-Lipschitz函数的一些广义微分性质

(h，φ)-凸函数与（h，φ）-Lipschitz函数的一些广义微分性质

摘要: 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Rⁿ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Rⁿ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Rⁿ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.

Abstract: 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.