Specifically, I am working to optimise something with the form $L(x,y) = g(y) + \frac{f(x)}{y^2}$, on $\mathbb{R}\times \mathbb{R}^*_+$.
The function $f$ is convex. The function $g$ is such that $y \mapsto g(y) y^2$ is convex as well. I am tempted to find $argmin(L')$, for $L'(x,y) := y^2 L(x,y)$. However I don't know if this would yield the same $argmin$ as the original problem. I would like to optimise $L'$ instead of $L$ because it would be a far, far easier problem to solve and my program has to run this optimisation on a heavy throughput of data in real time. Any help would be greatly appreciated.