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Given a symmetric positive semi-definite matrix $\mathbf Y$ and a convex set $\mathcal M$ which is a subset of all symmetric positive semi-definite matrices (consider a simple case of $\mathcal M$: a line segment between two symmetric positive semi-definite matrices). The optimization problem is as follows: \begin{equation} \min\limits_{\mathbf V} \max\limits_{\mathbf X\in \mathcal M} \frac{tr(\mathbf V^T \mathbf X \mathbf V)}{tr(\mathbf V^T \mathbf Y \mathbf V)} \end{equation}

I know it is not the best choice to iteratively solve the minmization problem and the maximization problem as it may not converge. Do you have any idea for this problem?

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    $\begingroup$ I think it would be more appropriate to ask this question in StackExchange. $\endgroup$
    – Pait
    Jan 8, 2014 at 14:39
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    $\begingroup$ The answer would depend on the convex set $M$ and on how it is represented. $\endgroup$
    – Michael
    Jan 9, 2014 at 0:39
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    $\begingroup$ What are the constraints on $V$? $\endgroup$ Jan 9, 2014 at 2:22
  • $\begingroup$ suppose $\mathcal M$ is a convex combination of several symmetric positive semi-definite matrices, $\mathbf V$ is a rectangular matrix without any particular constraint. $\endgroup$
    – Lucifer010
    Jan 13, 2014 at 0:55

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