Robust Rayleigh quotient minimization and nonlinear eigenvalue problems

Zhaojun Bai, Ding Lu, Bart Vandereycken

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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.

Original languageEnglish
Pages (from-to)A3495-A3522
JournalSIAM Journal on Scientific Computing
Issue number5
StatePublished - 2018

Bibliographical note

Publisher Copyright:
© 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


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