Given a generic linear programming problem i.e.
- minimize $\hat C^T \hat X$
- subject to $\hat A \hat X \leq \hat B$
- and $\hat X \geq 0$
Is it guaranteed (mathematically speaking) that a solution exists? I'm inclined to say yes (just as a raw guess withouth any thought), but is there a proof of this?
The link here lays out the requirements for the optimal solution to exist. If the constraint region is convex and nonempty than we are guaranteed to find a solution at one of the vertices. The convexity of constraint region is key for the solution, so the solution for your setup will always exist when $\hat{A}\hat{X}=\hat{B}$ has non-negative solutions.
EDIT: There exist some cases when the feasible region is open, and in those cases a solution does not exist because of unboundedness (especially for cases when $\hat{A}\hat{X}>\hat{B}$. A nice discussion about the unique solution of LP can be found here