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


*

*Orbitopes 

*Group majorization and a host of majorization inequalities induced by groups (which we may broadly view as being objects in optimization)

*Optimization over covariance matrices that exploits some group theory.
There are certainly more examples out there, but these should help you get started.
A: 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 $f$ !
