## Array Processing Convexity Problem

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?

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The problem as stated is not convex, even in the case K=1, because of the constraint which has b on the surface of a sphere. Actually any b other than those making the denominators 0 could be allowed, but that's still not a convex set. And since your objective is homogeneous of degree 0 in b, it's not going to be concave or convex on convex subsets of the domain except in trivial cases. – Robert Israel Mar 31 2011 at 16:50