预不变凸函数的一个等价条件

An Equivalent Condition of Preinvex Function

广义凸性在数学规划与最优化理论中具有十分重要的作用。本文通过将对多元实值函数的研究转化为对单变量的实值函数的研究,首先证明了当X为关于η的不变凸集,η满足条件C,f满足条件D时,对任意给定的x,y∈X,λ∈[0,1],F(λ)=f(y+λη(x,y))是凸函数当且仅当f为关于η的预不变凸函数。在此基础上建立了二次连续可微的预不变凸函数的一个等价条件:设X为关于η的开不变凸集,η满足条件C,f二次连续可微且满足条件D,则f关于η为预不变凸函数等价于x,y∈X,η(x,y)T▽2f(x)η(x,y)≥0。本文的结果为判断函数的预不变凸性提供了新的思路。

Generalized convexity has been playing an important role in mathematical programming. In this paper, an equivalent condition of twice continuously differentiable preinvex function is established by transforming multivariate real-valued function into univariate real-valued function. Suppose that X be open invex set with respect to η, ηsatisfies condition C, f be twice continuously differentiable and satisfies condition D. Then f is preinvex function with respect to η if and only if arbitary x,y∈X, η（x,y） ^T ↓△^2f（ x ） η （ x, y） ≥ 0. Our resuits provide new thoughts to verify the preinvexity of function and also generalize some known results.