I'm investigating optimal solutions to equations of the form:
Subject to the constraint:
and are also constrained independently of to keep the expression real, but I don't think this is important.
This gives the Lagrangian:
So taking the derivative with respect to gives:
Is this correct so far? Because it doesn't seem like this series of equations has a solution in general.
Oddly, it also seems like making the change of variables does give a solution. Following the above method, the equations are:
Which can be solved by setting all but one to 0, and the remaining one to 1.
Am I making some mistake here?