Optimization over unit vectors

Question

I am trying to optimize a certain nonlinear function over real or complex vectors with unit $L^2$ norm, i.e., I am trying to maximize a function $|f(x_1,\ldots,x_n)|^2$ subject to $\sum_i |x_i|^2 =1$. I am wondering if this goes by a special name or if it can be mapped to some well known (convex?) optimization problem, or if there are efficient numerical methods to deal with the optimization constraint.

Answer 1

Without any information on $f(.)$ it's impossible to say something. However, I don't think you can find a way to write this problem in convex form: first you are maximizing a quadratic function of $f(.)$ and this is already a bad sign. Moreover the domain $\| x \| = 1$ is nonconvex. A trick that sometimes can be used to optimize over unitary vectors is to relax the constraint to $\| x \| \leq 1$, and then add to the cost function some monotonic function of the norm of $x$, in such a way that $x$ is pushed to the boundary of the unit circle. This however is very likely to be in contrast with the cost function you actually want to maximize...