Robust Rayleigh quotient minimization by solving nonlinear eigenvalue problems (NEPv)


These are MATLAB implementations of the NEPv approach [1] for the robust Rayleigh quotient minimization

\[ \rho(z)= \max_{\substack{\mu\in \Omega\\ \xi\in \Gamma}} \frac{z^TA(\mu)z}{z^TB(\xi)z} \qquad \xrightarrow[]{z\neq 0} \qquad \min \]


\[ A: \quad \mu\in\Omega\ \mapsto\ A(\mu)\in \mathbb R^{n\times n } \quad\text{and}\quad B: \quad \xi\in\Gamma\ \mapsto\ B(\xi)\in \mathbb R^{n\times n } \]

are analytic matrix valued functions, with \(A(\mu)\succ 0\) and \(B(\xi)\succ 0\) for all \(\mu\in \Omega\) and \(\xi\in \Gamma\), and \(\Omega\subset \mathbb R^m\) and \(\Gamma\subset \mathbb R^p\) are compact. By solving the maximization problem explicitly, we can write the objective function as a nonlinear Rayleigh quotient

\[ \rho(z) = \frac{z^TG(z)z}{z^TH(z)z}. \]

The minimizer of \(\rho(z)\) can then be computed by the NEPv characterization [1] of the problem.

This package contains two NEPv solvers, together with example data and routines used in the paper [1]. Those provided examples consist two robust RQ applications, namely, the robust GEC and robust CSP.

To apply the NEPv solvers for other applications, users should provide the matrix-valued functions \(G(z)\) and \(H(z)\), and/or their (1st) derivatives. We also assume the minimizer \(z_*\) is a smooth point of \(\rho(z)\), otherwise convergence of the algorithms is not guaranteed.



  1. Robust Rayleigh quotient minimization and nonlinear eigenvalue problems
    with Zhaojun Bai and Bart Vandereycken, to appear in SIAM J. Sci. Comput., 2018. (preprint)