对称矩阵部分特征值之和的非光滑分析

NONSMOOTH ANALYSI OF THE SUMS OF EIGENVXLUES OF A SYMMETRIC MATRIX

本文研究n阶实对称矩阵A的前m项最大特征值之和fm（A）的非光滑分析问题．利用Ky－Fan的关于特征值之和的变分原理以及凸分析理论，得到了fm（A）的次梯度和方向导数的显式表达式．

Let A be an n- by-n symmetric matrix. We study in this paper a problemof nonsmooth analysis of fm(A), defined as the sum of the m largest eigenvalues of A. Basingon the Ky-Fan's variational principle for fm(A) and the techniques from convex analysis, wederive explicit formulas for the subdifferential as well as the directional derivative of the sumfm(A) viewed as a function of A.