凸函数性质的推广

Extension of Properties of the Multivariate Convex Function

凸函数理论是数学中相对年轻的一个分支,随着数学规划,对策论和最优控制理论等学科发展的需要,凸分析日益受到人们的重视。对一元函数的凸性研究已有丰富的成果,但对多元凸函数的研究不是很多。文章主要对一元凸函数的定义及其性质进行了推广。给出多元凸函数定义,凸凹性判别定理,判别定理是由多元函数的微分学基础上给出的,根据判别定理可以判断n元c1类和c2类函数的凸凹性,在多元凸函数的定义和判别定理的基础上我们把一元凸函数的有些性质推广到n元凸函数上。

Convex function theory is a relatively young branch in mathematics. Since the development of Mathematical programming ,game theory, optimal control theory and other subjects, convex analysis has been attached more importance by people increasingly.. The research on the convex function of one variable has gotten abundant achievements, but not for the one with multiple variable. In this paper, we extend mainly on the definition and nature of convex function of one variable. We give the definition of Convex Function with multiple variables, the discriminate theorem of convex and concave. The theory is given based on the differential calculus of multiple variable. According to the theorem, we can determine the convex and concave of c^1 function and c^2 function with n variables. On the basis of the definition and Determinate theorem, we extend some properties of convex function of one variable to the one of multiple variables and give some applications of convex function.