I have a constrained maximization problem (maximizing a functional), with number of constraints being uncountable infinite.

It looks something like this. I want to maximize the convex functional $C(f)$ over $f \in S$ with the constraint that look like $G(f,\phi) = 0 \forall \phi \in C^{\infty}(\Omega)$. Clearly these constraints are infinite in number (infact uncountably infinite).

I don't know how to use Lagrange multiplier technique in this context. Hence I request for a reference for a theory in this regard. I am not aware of any such concepts and no idea on what terminolgy I should search on google.

Edit : The example cited may not represent a typical case, neverthless I want a reference to the concerned generic theory.

Optimization by Vector Space Methodsby David Luenberger is a good place to start. $\endgroup$ – Christian Clason May 12 at 8:45