How to solve the following minimax problem quickly? The variables are all continuous.
$$\max_{x_{1}, x_{4}, x_{5}} \min_{x_2,x_3} \vec{c}^{\intercal} \vec{x}$$
subject to the following constraints: $$A\vec{x} \ge \vec{b}$$ $$\vec{x} \ge 0$$
where
$$A = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0.4 & 0 \\ 0 & 0 & -1 & 0 & 0.4 \\ 1 & 0 & 0 & -1 & -1 \\ 0 & -0.6 & 0 & 1 & 0 \\ 0 & 0 & -0.6 & 0 & 1 \end{bmatrix}$$
$$\vec{b}^{\intercal} = \begin{bmatrix} 1 & -1 & 2 & -2 & 1 & 1 & -1 & 0.6 & 0.6 \end{bmatrix}$$
$$\vec{c}^{\intercal} = \begin{bmatrix} 1 & -0.6 & -0.6 & 0 & 0 \end{bmatrix}$$
I tried to split the variables into two groups $x_1, x_4, x_5$ and $x_2, x_3$. And replace the inner minimization by its dual, which is a maximization. But that makes the objective function quadratic instead of linear:
$$ \max_{x_1, x_4, x_5} \left( \max_{x_2, x_3} \vec{b}^{\intercal} \vec{y} - \vec{x}^{\intercal}A_{1,4,5}^{\intercal} \vec{y} \right)$$ where $A_{1,4,5}$ is same as $A$ except columns 2 and 3 are all zeros.