r-预不变凸函数的一个等价条件

An Equivalent Condition of r-preinvex Functions

在上半连续条件给出了r-预不变凸函数一个等价条件。本文利用上半连续函数在紧集上必有最大值,设K是关于η的开不变凸集,η满足条件C,f上半连续且满足f(y+η(x,y))≤f(x),x,y∈K,得到f关于η为r-预不变凸函数当且仅当■α∈(0,1),x,y∈K,s.t.{f(y+αη(x,y))≤log(αerf(x))+(1-α)erf(y))1r,r≠0f(y+αη(x,y))≤αf(x)+(1-α)f(y),r=0。本文排除了K是开集这一条件,并且没用A在[0,1]上的稠密性。

An equivalent condition of r-preinvexity was given by condition C which was introduced by S. R. Mohan and S.K. Neogy. In this paper another proof is provided about the equivalent condition by the use of the conclusion which upper semicontinuous function has maximum on compact set. That is, let K be an open invex set with respect to η and η satisfies condition C. Let f be a upper semicontinuous function that satisfies f（y＋η（x,y））≤f（x）,A↓x,y∈K Then f is a r-preinvex function for the same η if and only if E←α∈（0,1）,A↓x,y∈K,s.t.{f（y＋αη（x,y））≤log（αe^rf（x））＋（1-α）e^rf（y）^1/r,r≠0 f（y＋αη（x,y））≤αf（x）＋（1-α）f（y）,r=0 The proof is absence of the assumptionwhich set K is open and A is dense on [0,1 ].