Given is the linear programming problem $(P) \max \left\{\left\langle c,x \right\rangle\mid A \cdot x \leq b, x \geq 0_n\right\}$ and its dual $(D) \min\left\{\left\langle b,y \right\rangle \mid A^T \cdot y \geq c, \,\, b \geq 0_m\right\}$
If $(P)$ is infeasible because its objective function can increase unrestricted, then what value can the objective function of its dual $(D)$ have? Proof.
I just read this article here https://en.wikipedia.org/wiki/Duality_gap and as it seems, there is no duality gap in linear programming. So this means that the solution of the objective function of $(D)$ will have the same as $(P)$ e.g. it will have value $\infty$ and be unbounded. I'm not sure about that but I think that's what I got from the article is correct but even then how is it possible to prove that?
If $(P)$ is infeasible, then there exists a $w$ such that $w \geq 0$ and $A^Tw = 0$ and $b^Tw <0$
Assume that the dual is feasible and $z$ is some dual feasible point then we have that $t+tw \geq 0$ and $A^T(z+tw)+c=0$ for all $t \geq 0$
This means that $z+tw$ is dual feasible for all $t \geq 0$ and because $t \rightarrow \infty$ we have $-b^T(z+tw) = -b^Tz-tb^Tw \rightarrow + \infty$ which means that dual is unbounded so value of objective function of $(D)$ is the same as of $(P)$..?
I hope you can tell me if this is correct like that or how to do it correctly?