We study the robust Rayleigh quotient optimization problem where the data matrices of the Rayleigh quotient are subject to uncertainties. We propose to solve such a problem by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). For solving the NEPv, we show that a commonly used iterative method can be divergent due to a wrong ordering of the eigenvalues. Two strategies are introduced to address this issue: a spectral transformation based on nonlinear shifting and a reformulation using second-order derivatives. Numerical experiments for applications in robust generalized eigenvalue classification, robust common spatial pattern analysis, and robust linear discriminant analysis demonstrate the effectiveness of the proposed approaches.
|Journal||SIAM Journal on Scientific Computing|
|State||Published - 2018|
Bibliographical noteFunding Information:
∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section January 29, 2018; accepted for publication (in revised form) August 1, 2018; published electronically October 18, 2018. http://www.siam.org/journals/sisc/40-5/M116768.html Funding: The first author was supported in part by NSF grants DMS-1522697 and CCF-1527091, and the second author by SNSF project 169115. †Department of Computer Science and Department of Mathematics, University of California, Davis, CA 95616 (firstname.lastname@example.org). ‡Department of Mathematics, University of Geneva, CH-1211 Geneva, Switzerland (Ding.Lu@ unige.ch, Bart.Vandereycken@unige.ch).
The first author was supported in part by NSF grants DMS-1522697 and CCF-1527091, and the second author by SNSF project 169115.
© 2018 Society for Industrial and Applied Mathematics.
- Nonlinear eigenvalue problems
- Rayleigh quotient
- Robust optimization
- Self-consistent-field iteration
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics