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An Equivalent Condition of Preinvex Function

Keywords: generalized convexity , preinvex function , twice continuously differentiable function , equivalent condition

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Abstract:

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,ηsatisfies condition C, f be twice continuously differentible and satisfiles condition D. then f is preinvex function with respect to η if and only if νx,y∈X,η(x,y)tV2f(x)η(x,y)≥0. Our results provide new thoughts to verify the preinvexity of function and also generalize some known results.

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