Consider the following (Lp). \begin{align}\min&\quad\mathbf{c} \cdot \mathbf{x}\\ \text{s.t. }&\quad A\mathbf{x} \geq \mathbf{b} \\ &\quad\mathbf{x} \geq \mathbf{0}\end{align}
If $S$ is the feasible set of (Lp), Prove that the following definitions are equivalent.
(Lp) is unbounded if $\; ∃ \{x^v\}_{v}⊆ S $ $ ;cx^v → −∞ \;$ as $v → ∞.$
(Lp) is unbounded if $\; ∀x ∈ S,\; ∃ \,y ∈ S\;;\; c y < cx.$
Proving 1 $\Rightarrow$ 2 is an easy result from the definition of limits.
Proving 2 $\Rightarrow$ 1 is not easy! There is no guarantee that you can find a sequence whose limit goes to $−∞$. The fact that we have a decreasing sequence of $cx$ for $x ∈ S$ does not necessarily imply (1). Maybe this sequence doesn't get less than a specific number, i.e, it has an infimum in $\Bbb R$.
Remark : If all coordinates of $c$ are positive, since we know that $ x \ge 0$ it results that for any $x ∈ S$, $cx \ge 0.$ In this case of course the limit of any such sequence is not $−∞$.
What are your thoughts? Are these 2 definitions really equivalent?
