r-预不变凸函数的一个性质(英文)

A Characterization for r-Preinvex Function

首先建立了一类r-预不变凸函数的一个等价条件,利用该等价条件给出了二次连续可微的r-预不变凸函数的一个性质;在适当的假设下,证明了如下结果:设XRn是关于向量值函数的开不变凸集,η满足条件C,f:X→R是二次连续可微的函数且满足条件D.则f是关于η的r-预不变凸函数当且仅当对任意的x,y∈X,r[f(y)T(x,y)]2+η(x,y)T 2f(y)η(x,y)≥0。本文的主要结果推广并改进了一些已有的主要结论。

In this paper,an equivalent condition for a class of r-preinvex function is established.A characterization for a twice continuously dierentiable r-preinvex function is obtained by the equivalent condition.Under some suitable conditions,the following result has been proved： Let XRn be open invex set with respect to η：X×X→Rn and η satisfy condition C.f defined on X is twice continuously dierentiable and satisfies condition D.Then f is r-preinvex function with respect to η if and only if x,y∈X,r[f（y）T（x,y）]2＋η（x,y）T2f（y）η（x,y）≥0.Our results improve and generalize some known results.