# Discrete mathematics – Given LP and its dual. If LP is infeasible, what is the objective value of its dual?

Given the problem of linear programming $$(P) max left { left langle c, x right row mid A cdot x leq b, x geq 0_n right$$ and his double $$(D) min left { left langle b, y right rangle mid A ^ T cdot y geq c, , , , b geq 0_m right$$

Yes $$(P)$$ is unrealizable because its objective function can increase without restriction, so what value can the objective function of its $$(D)$$ to have? Evidence.

and as it seems, there is no difference in duality in linear programming. It therefore means that the solution of the objective function of $$(D)$$ will have the same as $$(P)$$ for example. it will have value $$infty$$ and be limitless. I'm not sure of that, but I think that's what I understood from the article, but even then, how is it possible to prove it?

Yes $$(P)$$ is infeasible, then there is a $$w$$ such as $$w geq 0$$ and $$A ^ Tw = 0$$ and $$b ^ Tw <0$$

Suppose that the dual is feasible and $$z$$ is a possible double point so we have that $$t + tw geq 0$$ and $$A ^ T (z + tw) + c = 0$$ for everyone $$t geq 0$$

It means that $$z + tw$$ is double 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 the double is limitless so value of objective function of $$(D)$$ is the same as $$(P)$$..?

I hope you can tell me if it's okay like that or how to do it right?