I have $N$ equations:
$$R_i=\alpha_i\frac{T_i}{(x-x_i)^2+(y-y_i)^2}, i=1..N$$
$T_i>0$, $0 < \alpha_i < 1$ and so $R_i>0$.
$R_i$, $x_i$ and $y_i$ are known quantities; $x$, $y$, $T_i$ and $\alpha_i$ are unknowns and so I have $2+2N$ unknowns.
Is it possible to find the unknown quantities? How?
Looking at the problem as a 2D geolocation problem we have that there are $N$ radio stations, each station is at a 2D known location $(x_i,y_i)$ and transmits an unknown power $T_i$ (omnidirectional) which is attenuated by an unknown coefficient $\alpha_i$ and is received as $R_i$ by a receiving antenna located at the unknown location $(x,y)$.
I was thinking to solve the problem as an optimization like this one
$$\underset{x,y,\alpha_i,T_i}{\operatorname{argmin}} \sum_{i=1}^N (R_i[(x-x_i)^2+(y-y_i)^2]-\alpha_i T_i)^2$$
but I am not sure whether it is correct and/or meaningful.
No, it's not possible. You have more unknowns than equations. Even if you combine the $\alpha_iT_i$ into $N$ unknowns (since there's clearly no hope of finding them individually), you can still choose $x$ and $y$ arbitrarily and then calculate the $\alpha_iT_i$ from the equations, so the system is underdetermined.