广义凸函数的结构理论

Structural Theory of Generalized Convex Functions

【目的】对广义凸函数的性质研究,特别是建立可微前提下广义凸函数与其梯度相关的等价刻画,为函数广义凸性的判别及它在规划方面的应用提供理论依据。【方法】利用已有相关结果,通过建立广义凸函数与一元凸(拟凸)函数的等价关系,进而将凸(拟凸)函数的性质移植给广义凸函数。【结果】指出了F凸(拟凸)函数与凸(拟凸)函数之间的等价关系,进而给出可微F凸(拟凸)函数的与梯度相关的等价刻画。最后,作为应用,利用所得结果给出GA-凸函数的若干判别方法。【结论】将凸分析的若干基本结果在广义凸函数方面做了统一推广,解决了可微F凸(拟凸)函数的判别问题。

[Purposes]The purposes of this article are give some properties of generalized convex functions and given the equivalent descriptions of generalized convex functions and their gradient base on the differentiable conditions.They are very important for the judgment of generalized convexity and the applications about programming.[Methods]We transplant the properties of convex（quasiconvex）functions to the generalized convex functions through establish the equivalent relationships between generalized convex and one-place convex（quasi-convex）functions.[Findings]we give the equivalent relationships between Fconvex（quasi-convex）and convex（quasi-convex）functions and give the equivalent descriptions of differentiable Fconvex（quasi-convex）functions about gradient.As the conclusions,we give some discriminate methods of GA-convex functions.[Conclusions]This article extends some conclusions of convex analysis at the generalized convex functions,so we can solve the discriminate questions about differentiable Fconvex（quasi-convex）functions.