Suppose these over-determined system of equations: $$ |\mathbf{x}^T\mathbf{v_n}| = A, \qquad n = 1,2,\cdots,N-1 $$
$$ \mathbf{v_n}= [1 \quad w^n \quad w^{2n} \quad \cdots \quad w^{(N-1)n}]^T , \qquad w = e^{-j\frac{2\pi}N} $$ where $P$ is a known support of positive-valued vector $\mathbf{x}$.
Since I failed finding analytic solution for this system of equations, I am trying to solve it using numerical optimization methods such as Gradient Descent, Newton, ... . First of all is it a good idea to do this minimization? $$ \min f(\mathbf{x}) = \sum_{n=1}^{N-1} \left||\mathbf{x}^T\mathbf{v_n}| - A\right|^2 $$ And which numerical algorithm would be better ? Also what method I should use to satisfy the constraints: $supp(\mathbf{x}) = P$ and $x_i \ge 0$ ?
Since accuracy is more important than speed for me, I think Newton would be better? and using gradient projection for constraints seems easiest one to use! What's your suggestions?