Converting a max-min problem to a max problem with a constraint

Question

The objective is to find the greatest lower bound of the variable $\mu$. The lower bound is resulting from the positive-semidefinite (PSD) constraint $$\tilde{\mathbf{T}}:=\left( \begin{array}{ccc} 16 \mu -x(2)-4 x(10) & \frac{x(8)}{2}+2 x(11)+x(12) & 8 \mu -x(2)-2 x(10)+x(12) \\ \frac{x(8)}{2}+2 x(11)+x(12) & 4 \mu -x(2)-2 x(5)-x(8)-2 x(10) & -x(2)+2 x(11)+x(12) \\ 8 \mu -x(2)-2 x(10)+x(12) & -x(2)+2 x(11)+x(12) & 2 (8 \mu -x(2)-2 x(10)+x(12)) \\ \end{array} \right)\succeq0,$$where $\mathbf{x}$ is a 12-dimensional vector. Now, I optimize over $\mathbf{x}$, when subjected to PSD conditions on 4 matrices $\mathbf{T}_i$ which depend linearly in the components of $\mathbf{x}$, and a set of given constants $\mathit{c}_j$. Symbolically, I write $\mathbf{T}_i(c_j,\mathbf{x})\succeq0$. Thus, the problem is $$\underset{\mathbf{x},\mathbf{T}_{i}\succeq0}{\mathrm{max}}\underset{\tilde{\mathbf{T}}\succeq0}{\mathrm{min}\ \mu}.$$ Does anyone have a idea how the internal minimization problem can be replaced with by additional constraint, say $c(\mu,\mathbf{x})$, for the other one? i.e., to receive $$\underset{\tilde{\mathbf{T}},\mathbf{x},\mathbf{T}_{i}\succeq0,c(\mu,\mathbf{x})=0}{\mathrm{min}\ \mu.}$$