一类用内积表达的函数凸性

A KIND OF FUNCTION CONVEX EXPRESSED MEANS OF INNER PRODUCT

本文研究用内积表达的某些函数,分析它们的凸性,其主要的结论如下:由一个实的半正定矩阵 An×n 及属于 n 维欧氏空间的向量 a,x 构成[x,Ax]+[a,x]+b(b 为常量)型关于向量 x 的一类内积函数,则此类函数为n 维欧氏空间上的凸函数。A,B 均为 n×n 阶实矩阵、a,x 均属于 n 维欧氏空间向量,那么由 Bx+a,A(Bx+a)构成关于 x 的内积函数,该函数在 n 维欧氏空间上凸的充分条件是 B为任意,A 为半正定的。

This paper studies some function that is expressed by means of inner product and analysis their convex. The hredominant result that followed: This kind of inner function on vector x in made of A_(n×n) which is a positive semi-definite matrix,vector α and x which are belong to n-dimension Euolidecm space.Then this kind of function is a convex function over n-dimen- sion Eulerian space. A and B are n×n real matrix,a and x are vector which are belong to n-dimension Enlerian space.Then this kind of inner function on x which is made of Bx+α,A(Bx+α) is a convex function over n-dimension Eulerian space if B is any matrix and A is positive semi-definite matrix