Consider the convex optimization problem $$ \min_{x \in R^n} ~f(x), $$ for a convex function $f : R^n \to R$.
Now let $g : R^n \to R^n$ be a smooth, bijective map with smooth inverse $g^{-1}$. Then we can optimize the problem $$ \min_{y \in R^n} ~f(g^{-1}(y)). $$ This problem is not always convex in y. However, I suspect that every critical point of $\min_y h(y)$ with $h = f \circ g^{-1}$ corresponds actually to a global minimum of the convex problem. It appears simple, but I have trouble showing this. Any help is appreciated.
I would also be interested in the constrained case, i.e., $x \in C$ for some convex set $C$.