Suppose we have a real function defined on a manifold $\cal{M}$. Is there any algorithm for finding all the critical points of this function? As a concrete example, consider the following function
$$ f = - \psi_1^* \psi_2 - \psi_2^* \psi_1 + g_1 |\psi_1|^4 + g_2 |\psi_2|^2 ,$$
where $g_1$, $g_2$ are two real positive numbers. The manifold $\cal{M}$ is defined as
$$ |\psi_1|^2 +|\psi_2 |^2 = 1. $$
Note that for this particular problem, the function $f$ is invariant under the transform $ \psi_{1,2} \rightarrow e^{i\theta }\psi_{1,2}$. We of course do not count this symmetry.
If this small problem can be solved, how about this bigger problem
$$ f = - \sum_{j=1}^N (\psi_j^* \psi_{j+1} + \psi_{j+1}^* \psi_j) + \sum_{j=1}^N g_j |\psi_j|^4 . $$
Here it is understood that $\psi_1 = \psi_{N+1}$.