I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal solution to satisfy certain properties. The goal is to find all such functions which in the optimal solution satisfy this condition I want. Said another way, find me a class of functions which when optimized subject to a set of constraints, gives me a solution satisfying some property.
How do I approach this problem? I can write the KKT conditions for this, but it doesn't get me anywhere. Someone said this may fall in the calculus of variations domain, but I don't know how to use that. What's a good starting point?