During my research in array processing I encountered this optimization problem:
\begin{aligned} & \underset{\mathbf{b}}{\text{maximize}} & & \sum_{k = 1}^K \frac{\mathbf{b}^H\mathbf{A}_k\mathbf{b}}{\mathbf{b}^H\mathbf{B}_k\mathbf{b}} \ & \text{subject to} & & ||\mathbf{b}||_2^2 = \alpha, \end{aligned}
Where $\mathbf{A}_k, \mathbf{B}_k \quad k \in 1,\ldots , K$ are semidefinite positive. When $K = 1$ the solution is the generalized eigenvector associated to the maximum generalized eigenvalue of the matrix pencil $(\mathbf{A}_1, \mathbf{B}_1)$ scaled in order to satisfy the norm constraint.
Problems arise when $K>1$.
In that case, Is the problem convex?
