Continuous minimax with linear objective and constraints

Question

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.

Answer 1

The problem is infeasible:

$(c_1,c_2) \Rightarrow x_1=1$

$(c_3,c_4) \Rightarrow x_2+x_3=2$

$(c_1,c_2,c_7) \Rightarrow x_4 + x_5 \le 2$

$(c_5,c_6) \Rightarrow -(x_2+x_3)+0.4(x_4+x_5) \ge 2 \text{ and so } (c_3,c_4,c_5,c_6)\Rightarrow x_4 + x_5 \ge 10$

The last two lines are not consistent. Note that we also obtain a conflict with $(c_8,c_9)$:

$(c_8,c_9) \Rightarrow -0.6(x_2+x_3)+(x_4+x_5) \ge 1.2 \text{ and so } (c_3,c_4,c_8,c_9)\Rightarrow x_4 + x_5 \ge 2.4$