Using Weierstrass theorem to prove having a finite optimal solution I was wondering if anyone has used Weierstrass theorem to prove that we have a finite optimal solution or has any reference for this claim.
I read in a paper that if an objective function is convex a, using Weierstrass theorem, we can conclude that the optimal objective function is finite, and the optimal set is non-empty. However, they have not provided a reference for that.
Thanks a lot for your help in advance,
 A: The Weierstrass extreme value theorem asserts that if you minimize a continuous function over a closed and bounded set in $\mathbb R^{n}$, then the minimum will be achieved at some point in the set.  

I read in a paper that if an objective function is convex a, using
  Weierstrass theorem, we can conclude that the optimal objective
  function is finite, and the optimal set is non-empty. However, they
  have not provided a reference for that.

To use the version of the Weierstrass theorem that I summarized above, you need the set under consideration to be closed and bounded and the function to be continuous.  On a subset of $\mathbb R^{n}$, if $f$ is convex, then it is also continuous.  However, there's nothing in your statement that says that the set of points is closed and bounded.  
It's easy to construct examples of convex functions on sets that are either not closed or not bounded where no minimum is achieved.  For example, minimize $f(x)=e^{x}$ on $\mathbb R$. 
The OP has now cited the paper that they were reading.  In this paper, a convex function $f$ is being minimized over a closed and bounded (compact) set $X$ in $\mathbb R^{n}$, so the extreme value theorem applies.
Statements and proofs of the Weierstrass extreme value can be found in many undergraduate analysis textbooks. See for example theorem 4.16 in the third (1976) edition of Rudin's Principles of Mathematical Analysis.
