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

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

where

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

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.

Link rbstrq.tar.gz (updated on

**Feb. 1, 2018**).

**Robust Rayleigh quotient minimization and nonlinear eigenvalue problems**

with Zhaojun Bai and Bart Vandereycken, to appear in*SIAM J. Sci. Comput.*, 2018. (preprint)

Email: Ding.Lu@unige.ch

Homepage: http://www.unige.ch/~dlu