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.


I just read this article here https://en.wikipedia.org/wiki/Duality_gap
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?