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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?

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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 $\min\cos(x)$.

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  • $\begingroup$ OK, that makes sense. The only thing I am struggling with then is what unbounded actually. Could you kindly elaborate on your point? $\endgroup$ Mar 13, 2021 at 14:13
  • $\begingroup$ 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? $\endgroup$ Mar 13, 2021 at 14:15
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    $\begingroup$ Does this clarify what unbounded/bounded functions are? $\endgroup$
    – DMcMor
    Mar 13, 2021 at 14:55
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    $\begingroup$ Yes, absolutely. Thanks for the help :) $\endgroup$ Mar 13, 2021 at 14:57

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