Consider a LP
$$\min_{x, y} c^Tx + d^T y$$ $$s.t. Ax + By \geq b$$ $$x, y \geq 0,$$
where $A$ is a square matrix such that $A\geq I$ componentwise.
Suppose that this LP is feasible. What is the minimum value of the objective function.
If at least one element of $c$ is negative, then the minimum value is $-\infty$.
If $c \geq 0$ componentwise, some elements of $d$ are negative and $\exists \hat{y} \geq 0$, $y_i = \infty$, $d_i < 0$ and $Ax + B\hat{y} \geq b$, then the minimum value is $-\infty$.
If $c \geq 0$ componentwise and the problem is bounded, then the KKT conditions of this problem are: $$c - A^T\lambda - \mu_1 = 0 \tag{1}$$ $$d - B^T\lambda - \mu_2 = 0 \tag{2}$$ $$\lambda^T(b - Ax - By) = 0 \tag{3}$$ $$\mu_1 x = 0 \tag{4}$$ $$\mu_2 y = 0 \tag{5}$$ $$Ax + By \geq b \tag{6}$$ $$x, y, \lambda, \mu_1, \mu_2 \geq 0 \tag{7}$$
But I still cannot solve the problem or just find the optimal value since I have no idea about matrix $B$ and $A$.
Also I am not sure how to find all of conditions such that this problem is unbounded.