Nonsmooth analysis of singular values. Part I: Theory

被引：95
作者：
Lewis, A [1 ]
Sendov, H
机构：
[1] Univ Waterloo, Dept Combinator & Optimizat, Waterloo, ON N2L 3G1, Canada
[2] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada
来源：
SET-VALUED ANALYSIS | 2005年 / 13卷 / 03期
基金：
加拿大自然科学与工程研究理事会;
关键词：
nonsmooth analysis; singular values; regular subdifferential; limiting; subdifferential; horizon subdifferential; von Neumann trace inequality; simultaneous diagonalization;
D O I：
10.1007/s11228-004-7197-7
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumann's trace inequality for singular values.
引用
收藏
页码：213 / 241
页数：29
相关论文
共 23 条
[1]  
BRICKELL F, 1970, DIFFERENTIAL MANIFOL
[2]  
Clarke F., 1998, NONSMOOTH ANAL CONTR
[3]  
Clarke FH, 1983, OPTIMIZATION NONSMOO
[4]  
Clarke FH, 1973, THESIS U WASHINGTON
[5]  
Gibson P. M., 1974, Linear Algebra and Its Applications, V9, P45, DOI 10.1016/0024-3795(74)90025-1
[6]  
Hiriart-Urruty J.-B., 2013, Convex Analysis and Minimization Algorithms I: Fundamentals, V305
[7]  
Horn R. A., 1986, Matrix analysis
[8]  
Horn RA., 2013, Topics in Matrix Analysis, V2nd ed, DOI DOI 10.1017/CBO9780511840371
[9]   Nonsmooth analysis of singular values. Part II: Applications [J].
Lewis, A ;
Sendov, H .
SET-VALUED ANALYSIS, 2005, 13 (03) :243-264
[10]   Group invariance and convex matrix analysis [J].
Lewis, AS .
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 1996, 17 (04) :927-949
123