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