# RobustRQ

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

## Description

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.