I want to maximize the sum of Rayleigh quotients:

$$\max_x\sum_{i=1}^n\frac{x^\top A_i x}{x^\top B_i x}$$

where $A_i$ and $B_i$ is positive definite. I've found a similar question here: minimization problem: sum of Rayleigh quotients. But it only deals with the case $n=2$, and therefore it seems that the methods there are not applicable for my problem.

Is there any suggestion for this problem? Does gradient descent/ascent work for this problem?


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    $\begingroup$ Since the sum of convex function is convex, I think that the same methods apply here. $\endgroup$ – nicoguaro Oct 22 '14 at 22:47
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    $\begingroup$ @nicoguaro: Note that this problem is a maximization problem, so you really want the objective function to be concave. Convexity won't help here. If it were a minimization problem, the property you cite would be useful. $\endgroup$ – Geoff Oxberry Oct 23 '14 at 18:50
  • $\begingroup$ @GeoffOxberry, sorry, my bad. $\endgroup$ – nicoguaro Oct 23 '14 at 18:58

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