预不变拟凸函数的一个充分条件

A Sufficient Condition of Prequasi-invex Function

对于可微的函数,其二阶导数可以刻画函数的凸性。受这种思想的启发,邢志栋等人根据微分方程的极值原理给出了拟凸函数的一个充分条件,本文利用文献[1]中建立的定理1,给出了二次可微的预不变拟凸函数的一个充分条件。X关于η(x,y)为不变凸集,二次连续可微函数f(x)满足条件D,η(x,y)满足条件C且η(x,y)下有界,若x∈X,▽2f(x)+g(x)▽f(x)T是半正定的(其中g(x):X■Rn→Rn是下有界函数),则f(x)关于η(x,y)是预不变拟凸函数。本文的结论是对文献[2]中相应结论的推广。

Generalized convexity has playing an important role in mathematical programming and optimization theory. In recent years many authors have been doing further research into generalized convexity and the applications in optimization theory and making a series of important conclusions. It is known that the second derivative can characterize the convexity of functions. Because of the enlighten- ment of this thoughts, Zhidong Xing and his fellowship gave a sufficient condition of twice differentiable quasiconvex functions by making use of the extremum principle about differential equation in reference [ 1 ]. In this paper a sufficient condition of twice differentiable prequasi-invex functions is constructed by making use of theorem 1 in reference [ 1 ]. The main results as follows ： Suppose the set X is invex with respect to η （x,y）, twice continuously differentiable function f（x） satisfies the condition D, η （x, y） satisfies condition C and is bounded below,for any x in X,△↓^2f（x）＋g（x）△↓f（x）^T is positive semi-definite g（x） is bounded below. Thenf（x） is prequasiiuvex functions. The conclusion improves and generalizes the corresponding result in reference [ 2 ].