I am in need of help with this problem. Suppose we have a $d\times d$ positive definite matrix $R$ whose diagonal entries are all $1$ (like a correlation matrix), denote the inverse matrix of $R$ by $S$. Consider the set (denoted by $C$) of all points $y\in\mathbb{R}^d$ satisfying the following equation: $$y^TSy=1.$$ The problem troubling me is, how can I maximize the minimum coordinate of points on $C$, namely I want to determine $$\max_{y\in C}(\min_{i\in\{1,2,...,d\}}y_i),$$ I know the range of searching can be confined to the subset $\{y\in C|y_i>0,\ \forall i\}$, and the case $d=2$ is not difficult since equations describing $C$ in this case are all in the form of $$y_1^2-2\rho y_1y_2+y_2^2=1-\rho^2,$$ where $\rho$ ranges in $(-1,1)$. The maximum must be attained at the only intersection of the line $y_2=y_1$ and $C$ in the first quadrant. But the case $d\geq 3$ seems complex and I have no idea till now.
Thanks for all your help.
It might help to reformulate as follows. Let $z$ represent the $\min$. Then the problem is to maximize $z$ subject to: \begin{align} z &\le y_i &&\text{for $i\in\{1,2,\dots,d\}$} \\ y^T S y &= 1 \end{align}