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 $f$ !