most recent 30 from http://mathoverflow.net 2013-05-21T10:35:49Z http://mathoverflow.net/feeds/question/60199 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60199/array-processing-convexity-problem mikitov 2011-03-31T15:48:04Z 2011-03-31T15:48:04Z

<p>During my research in array processing I encountered this optimization problem:</p> <p>\begin{aligned} &amp; \underset{\mathbf{b}}{\text{maximize}} &amp; &amp; \sum_{k = 1}^K \frac{\mathbf{b}^H\mathbf{A}_k\mathbf{b}}{\mathbf{b}^H\mathbf{B}_k\mathbf{b}} \ &amp; \text{subject to} &amp; &amp; ||\mathbf{b}||_2^2 = \alpha, \end{aligned}</p> <p>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.</p> <p>Problems arise when $K>1$. </p> <blockquote> <p>In that case, Is the problem convex? </p> </blockquote>