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?
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? 


Symmetry groups are certainly an issue in integer programming. Orbital branching is one way of dealing with it. Core points are another relevant concept. 


To some extent. Here's some relevant material where group theoretic objects show up in optimization (though a lot of it is convex algebraic geometry).
There are certainly more examples out there, but these should help you get started. 


Just an elementary remark, if the function $f:X\to\mathbb{R}$ is invariant under the action of a $G$ on $X$ (meaning that $f(g\cdot x)=f(x)$ then you can, at least morally, search for your minimum on the quotient space $X/G$, which is smaller. However this quotient might be not as nice as the space $X$ you started with. I'm not sure if this is directly used in some optimization algorithms, however, it can be used implicitly at the modelization step. For instance if you have a function $f$ on $\mathbb{R}^n$ which is invariant under the orthogonal group $O(n)$, then you know that $f(x)=g(\x\)$ for some function $g:\mathbb{R}_+\to\mathbb{R}$, and you'd better optimize $g$ on $\mathbb{R}_+$ instead of $f$ ! 

