I got a constrained optimisation problem with a single equality constraint. Assuming the input dimensions are A_n the constraint is sum(A_n) = A. The optimisation metric is sum(f_n(A_n)).

Now I used Lagrange to arrive at a bunch of
g_n(A_n) = lambda

f_n and g_n are quite complex but all f_n/g_n are the same respectively, just with different constants. Is there any general way to solve this or any approach you could recommend?

This is not homework by the way, just a private interest.