How to numerically solve for a variant of the weighted least squares

I encountered the following problem, and I don’t have a good idea on where to start. Suppose that we have the following weighted least square problem:
$$
hat{beta}_{WLS} = argmin_beta(y – Xbeta)^TW(y – Xbeta).
$$
Normally, this would have a closed form. But my question says that now $W$ is no longer a matrix with known diagnoal elements but rather $W_{ii} = w_i(alpha)$ for some unknown parameter $alpha$. How do I numerically solve the above least square estimates? Can somebody give some hint and point me to some sources? Thank you!