文件名称:
求解广义瑞利商的极值问题，包括局部最优值和全局最优值的论证
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文件大小: 2mb
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上传时间: 2020-04-25
详细说明：Given symmetric matrices B,D ∈ R n×n and a symmetric positive definite
matrix W ∈ R n×n , maximizingthe sum of the Rayleighquotientx ? Dx andthe gener-
alized Rayleigh quotient
x ? Bx
x ? Wx
on the unit sphere not only is of mathematical interest
in its own right, but also finds applications in practice. In this paper, we first present a
real world application arising from the sparse Fisher discriminant analysis. To tackle
this problem, our first effort is to characterize the local and global maxima by investi-
gating the optimality conditions. Our results reveal that finding the global solution is
closely related with a special extreme nonlinear eigenvalue problem, and in the spe-
cial case D = μW (μ > 0), the set of the global solutions is essentially an eigenspace
corresponding to the largest eigenvalue of a specially-defined matrix. The characteri-
zation of the global solution not only sheds some lights on the maximization problem,
but motives a starting point strategy to obtain the global maximizer for any monoton-
ically convergent iteration. Our second part then realizes the Riemannian trust-region
method of Absil, Baker and Gallivan (Found. Comput. Math. 7:303–330, 2007) into
a practical algorithm to solve this problem, which enjoys the nice convergence prop-
erties: global convergence and local superlinear convergence. Preliminary numerical
tests are carried out and empirical evaluation of its performance is reported.
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