# How to solve such an optimization problem efficiently？

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: $$\min\limits_{\mathbf V} \max\limits_{\mathbf X\in \mathcal M} \frac{tr(\mathbf V^T \mathbf X \mathbf V)}{tr(\mathbf V^T \mathbf Y \mathbf V)}$$

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 '14 at 14:39
The answer would depend on the convex set $M$ and on how it is represented. – Michael Jan 9 '14 at 0:39
What are the constraints on $V$? – Felix Goldberg Jan 9 '14 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 '14 at 0:55