I am new to linear programming. I came across a problem which seems easy but I am failing to figure out how to prove it.
Suppose the following LP is feasible:
$\text{max}$ $c^Tx$
$\text{s.t. }$ $Ax\leq b.$
Prove that the optimum objective value of this LP is bounded if and only if the associated LP shown below
$\text{max}$ $c^Tx$
$\text{s.t. }$ $Ax\leq 0,c^Tx\leq 1,$
has zero as its optimum objective value.
I am familiar with bounded and unbounded LP problems. But I did not find anything about bounded optimum value on the text that I am following. I recall several theorems dealing with unbounded LPs. I am also familiar with Farkas' lemma and duality.
If anyone can help me in understanding the above statement and supply a proof/proof sketch, it would be super helpful.
