Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation  

作　　者：ZHANG LeiHong LI RenCang 

机构地区：[1]Department of Applied Mathematics, Shanghai University of Finance and Economics [2]Department of Mathematics, University of Texas at Arlington

出　　处：《Science China Mathematics》2015年第7期1549-1566,共18页中国科学：数学（英文版）

基　　金：Acknowledgements The first author was supported by National Natural Science Foundation of China(Grant Nos.11101257 and 11371102);the Basic Academic Discipline Program,the 11th Five Year Plan of 211 Project for Shanghai University of Finance and Economics;supported by National Science Foundation of USA(Grant Nos.1115834and 1317330);a Research Gift Grant from Intel Corporation

摘　　要：The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr（VTAV）/tr（VTBV）＋tr（VTCV）over the Stiefel manifold{V∈Rm×l ｜VTV=Il}（l〈m）,naturally leads to a self-consistent-field（SCF）iteration for computing a maximizer.In this part,we analyze the global and local convergence of the SCF iteration,and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration.This is one of the advantages of the SCF iteration over optimization-based methods.Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.Abstract The necessary condition established in Part I of this paper for the global maximizers of the maximization problem max V tr(VTAV)/tr(VTBV)+tr(VTCV)over the Stiefel manifold{V∈Rm×l |VTV=Il}(l<m),naturally leads to a self-consistent-field(SCF)iteration for computing a maximizer.In this part,we analyze the global and local convergence of the SCF iteration,and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration.This is one of the advantages of the SCF iteration over optimization-based methods.Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.

关 键 词：trace ratio Rayleigh quotient Stiefel manifold nonlinear eigenvalue problem optimality condi-tion self-consistent-field iteration EIGENSPACE 

分 类 号：O175.8[理学—数学]