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?