I am working on the following exercise:
Consider the LP
$max \ c \cdot x$ with $Ax \le 0$ and $x \ge 0$
Show that either $x=0$ is an optimal solution or the LP is unbounded.
My idea for a proof goes as follows:
We assume that $0$ is not an optimal solution. Now for the sake of contradiction we suppose that $\overline{x}$ is an optimal solution. Now we need to show that we can find a feasible $y$ sucht that $c \cdot y \ge c \cdot x$, but I do not see how to do this. Could you help me?
Clearly $x=0$ is a solution. Assume that the maximum is finite, then it must be greater than $0$. Assume $\overline{x}$ is the optimal solution. Then for all $\lambda > 0$, $\lambda \overline{x}$ is also a solution but then $c\overline{x} \geq \lambda c\overline{x}$, but this can only be true if $c\overline{x} = 0$. Hence, $x=0$ is an optimal solution.