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).

1. Orbitopes
2. Group majorization and a host of majorization inequalities induced by groups (which we may broadly view as being objects in optimization)
3. Optimization over covariance matrices that exploits some group theory.

There are certainly more examples out there, but these should help you get started.