Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem \begin{align} \max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\ \mathbf{u}^H\mathbf{A}_2\mathbf{u}\geq 0 \end{align} I know how to numerically solve it, for example Semi-Definite Relaxation is easily applicable. I was wondering if there is an analytic approach to it. Note that without the constraint, this is equivalent to the maximum rayleigh-ritz ratio.

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem \begin{align} \max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\ \mathbf{u}^H\mathbf{A}_2\mathbf{u}\geq 0 \end{align} I know how to numerically solve it, for example Semi-Definite Relaxation is easily applicable. I was wondering if there is an analytic approach to it. Note that without the constraint, this is equivalent to the maximum Rayleigh-Ritz ratio.