Let the following weighted power sum system of equations on $x_j$, with $j= 1,2,\ldots, n$ :
$$\alpha_{11} \cdot x_1 + \alpha_{12}\cdot x_2 + \cdots + \alpha_{1n}\cdot x_n = b_1$$ $$\alpha_{21}\cdot x_1^2 + \alpha_{22}\cdot x_2^2 + \cdots + \alpha_{2n} \cdot x_n^2 = b_2$$ $$\vdots$$ $$\alpha_{m1}\cdot x_1^m + \alpha_{m2}\cdot x_2^m + \cdots + \alpha_{mn}\cdot x_n^m = b_m,$$
where $\alpha_{ij}, b_i, x_j \in \mathbb{C}$ and $m \geq n$. I am looking for a method that solves this system in the least-squares sense.
Any idea of how to tackle this problem?
Thanks in advance for any help!