Is group theory useful in any way to optimization? For what I have seen, optimization uses a lot of linear algebra and convex analysis, but I have not seen any group theory being used, so I was curious about it. 
Is group theory useful in any way to optimization?
 A: Yes, certainly. Please search for the papers with keywords "symmetry breaking constraint programming" to find out more information. In particular, see Symmetry in Constraint Programming by Ian P. Gent, Karen E. Petrie, Jean-François Puget, Chapter 10 in Handbook of Constraint Programming, Edited by F. Rossi, P. van Beek and T. Walsh, Elsevier, 2006.
Basically, knowing symmetries of solutions allows to reduce the search space: if it is known that the property of being a solution to the problem is preserved under some symmetries, then we have to check only representatives of orbits of solutions with respect to these symmetries, and groups naturally occur in studying this.
A: Yes, One of the famaus one is that Rubik's Cube can be solved by at most $20$ move.
Since symetries of Rubik's cube forms a group, it is proved by the means of properties of Cayley graph of its group.
http://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube
It has also some applications in crypto systems. (To broke the codes faster by brute force)
A: In the case of convex optimization, not really - you're playing with properties of convex functions and differentiability abilities (usually modifying the idea of gradient descent in deterministic or stochastic form).
However, for some types of combinatorial optimization, some group and field theory is useful (some arise in existence of certain types of codes, for example). A case where this is applied is in chemistry, where a lot of properties of molecules are encapsulated via symmetry groups and what not, so group theory provides some of the constraints for some function which needs to be optimized (such as energy of a configuration). There is also some interplay with algebraic geometry. 
