I am a functional analyst by training, but I am doing some numerical experiments which require me to minimise continuous functions $f:SO(3)\longrightarrow [0,+\infty)$ using a computer (I know that each $f$ will be continuous, but cannot safely assume anything better).

I have access to two Nelder-Mead implementations: one which works in $SO(3)$ directly (using slerp and Karcher means to modify the simplex), and one which requires me to supply a parameterisation through which $f$ can be pulled back to $\mathbb{R}^3$.

My question is really just this: is there any intrinsic problem with using Euler-angles or an exponential parametrisation and minimising pullback of $f$ to $\mathbb{R}^3$ instead of $f$ itself?

My intuitive feeling is that, since I don't know anything about $f$ anyway, composing with something else is unlikely to hurt as long as it doesn't overly restrict where my simplex ends up going. Working in $\mathbb{R}^3$ has the advantage of being computationally cheaper, but the amount of literature describing co-ordinate free approaches makes me question whether this it's really a sensible option.

Thanks in advance for any advice!