Can we say the primal problem get $+\infty/-\infty$ if the dual problem is infeasible?

Question

It can be easily verified that if the optimal value of the primal problem is $+\infty/-\infty$, the dual problem is infeasible. How about the inverse? If the dual problem is infeasible, can we say that the optimal value of the primal problem is $+\infty/-\infty$ if the primal problem is feasible?

Answer 1

It's possible for both the primal and dual problems to be infeasible. An example from Vasek Chvatal's LP textbook is:

$\max \;\; 2x_{1}-x_{2}$

$ x_{1}-x_{2} \leq 1$

$-x_{1} +x_{2} \leq -2$

$x_{1}, x_{2} \geq 0$.