I am currently tackling an optimization problem involving complex valued vectors. However the optimization is solely about finding the optimal “direction” of the vector. So any (complex-valued) scaling does not matter. To make the optimization procedure (currently gradient descent) faster and more robust I want to constraint the optimization to the complex valued unit hypersphere.
I would be interested whether there is something like complex-valued spherical coordinates so I can translate my problem into coordinates which have one degree of freedom less?
I would also be interested whether there is something like constraint gradient descent for my purpose (Maybe similar to Lagragian optimization, but in my case there is likely no closed-form solution) ?