多元凸函数的性质及其应用  

The Properties and Applications of Convex Function of Many Variables

作　　者：游煦[1] 

机构地区：[1]北京石油化工学院数理系,北京102617

出　　处：《北京石油化工学院学报》2008年第4期61-64,共4页Journal of Beijing Institute of Petrochemical Technology

摘　　要：从多元凸函数的定义及文献中已有的性质出发,利用方向导数和极限等数学工具,给出了一个判别多元函数凸性的充分必要条件,进一步利用函数f(x)的Hesse矩阵Hf(x)的半正定性来判定函数的凸性。特别地,对于二次函数f(x)=1/2xTAx+bTx直接利用矩阵A的正定性可以判别它的凸性。这在实际应用中有一定的意义。From the definition of convex function of many variables and the conclusion in reference[1,3], we can get some necessary and sufficient conditions for judging the convexity of function of many variables by directional derivative and limit. Furthermore, we obtain the method of the determination of convex function of many variables by using the positive semi-definite of the Hessian matrix of the function f（x）. At last, we derived the method of the determination of strict convex quadratic function f（x）=1/2x^TAx＋b^Tx with the positive definite matrix A.

关 键 词：多元凸函数 梯度向量 HESSE矩阵 半正定阵 二次函数 

分 类 号：O174[理学—数学]