If $g,f:X\rightarrow Y$ are functions and $g$ is monotone then
$$
\operatorname{argmin}_{x \in X} f(x) 
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$

So monotonicity is sufficient for preserving extrema.  Are necessary and sufficient conditions known for $g$ to preserve minima?  If so does anyone know a good reference?