Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to minimize
$$L(x) = (y - f(x))^2 = y^2 + f(x)^2 - 2yf(x).$$
In general $L$ will not be convex: for $y > 0$ and $f$ nonnegative the second term is convex and the last term is concave.
Is there any way to formulate such a reconstruction problem in a convex way?
Thanks a lot!