An optimal function $f(x)$ is sought that minimises some loss function $g(f(x))$. The determination of an optimal function $f(x)$ contrasts traditional optimisation problems where the function is known and the objective is find the global minimum or maximum value of that function. In this problem it is the shape of the function $f(x)$ that is to be optimised.
I suspect that there is already extensive literature on this problem however I am having difficulty locating it. The closest thing I can find is shape optimisation which seeks an optimal topology that minimises some function. However I am interested in the case where the domain is one-dimensional and the value of the function is continuous (not 0 or 1 in the case of topology). A simple approach to this problem is to discretise the function into a number of points and then accept perturbations if the perturbed function minimises the loss function $g(x)$ however I would like to read some literature on this type of problem that considers a number of different discretisation and optimisation methods.
I also found this stack exchange question: finding an optimal function subject to some constraints.
