True or false?
(a) Deleting a constraint leaves the feasible region larger.
(b) Adding a constraint leaves the feasible region either unchanged or smaller.
(c) An LP cannot have an unbounded objective function value.
Here are my answers:
(a) True because when I graphed a random problem, it looked like the feasible region got bigger.
(b) True -Same as (a) but saw it get smaller.
(c) True because it can have an optimal objective function value but probably not an unbounded objective function value.
But I don't understand how all three are true. Help is very much appreciated!!
Best Answer
$(A)$: It can also remains the same. For example for a $2$-dimensional problem, if $0 \le x\le 1$ and $0 \le y \le 1$ and $x+y \le 2$. Removing the last constraints doesn't change the feasible set.
$(B)$: Let $D$ be the initial feasible set and $D'$ be the new feasible set with more constraitns, then we have $D' \subseteq D$ since any element of $D'$ would satisfy the conditions to be in $D$.
$(C)$: Consider $\max x$, subject to $y = 0$. Is it bounded?