# What is the difference between an unbounded solution and infinite optimal solutions?

I am currently studying Linear Programming, and one of the few things that I haven't found a solution for (pun obviously intended) is the difference between a solution being either unbounded or having infinite optimal solutions.

My thinking is, if something is unbounded, is that not the same as it being infinite? I must not have a proper understanding of both, as I cannot see why my thinking would correct.

If possible, could someone illustrate the differences with a graph?

Unbounded Solution refers to one solution being unbounded, in the sense that the $$y = x$$ is unbounded.
Infinite Optimal Solutions refers to an infinite family of functions that are all optimal solutions, in the sense that $$x = \pi + 2n\pi$$ is an infinity family of solutions to the problem $$\min\cos(x)$$.
• In fact, I may understand it, but without a graph I won't be able to fully show, but I'll try: if you have two constraints and the solution is in between the two lines, that would be considered to have infinitely many solutions; whereas if you had one constraint that was greater than and $z$ is to be maximised, the optimal solution would be anywhere above that line, i.e. unbounded. Correct? Mar 13, 2021 at 14:15