Extension to SAFE screening rule for Lasso In El Ghaoui et al. (2010), "Safe feature elimination in sparse learning" and following works, screening rules are derived for Lasso (as well as other L1-penalized problems):
$ \min_w \|y-X w\|^2 + \lambda \|w\|_1$. 
The SAFE rule in particular guarantees $w_i = 0$ if $|X_i ^T y| < \lambda - \|X_i\|_2 \|y\|_2 \frac{\lambda_{max}-\lambda}{\lambda_{max}}$, where $\lambda_{max} = \|X^T y\|_\infty$.
Suppose instead I have different regularization penalties for different coefficients:
$ \min_w \|y-X w\|^2 + \|\lambda \circ w\|_1$. 
Is the corresponding SAFE rule now $|X_i ^T y| < \lambda_i - \|X_i\|_2 \|y\|_2 \frac{\lambda_{max}-\lambda_i}{\lambda_{max}}$?
 A: Consider a change of variables:
$$\min_{w'} \lVert y - X' w \rVert^2 + \lVert w' \rVert_1$$
where $w' = \lambda \circ w$ and $X'_{ij} = X_{ij} / \lambda_j$.
This problem is equivalent, and allows us to directly apply the original SAFE rule with $\lambda = 1$:
$$\lvert {X'_i}^T y \rvert < 1 - \lVert X' \rVert_2 \lVert y \rVert_2 \frac{\lambda_\max - 1}{\lambda_\max}$$
with $\lambda_\max = \lVert {X'}^T y \rVert_\infty$.
Note that it is not equivalent to your proposed rule, but is about as easy to compute.
It may be the case (by following through the original proof) that your proposed rule is also valid – it seems reasonable enough – but without doing that I'm not sure.
One drawback of this, relative to your proposed rule, is that all the quantities must be recomputed if you change the relative contributions of $\lambda$. (A global scaling can be easily handled by adding a \lambda_0 back into the reformulated problem.) The advantage is that it certainly works, unlike your proposal which may or may not be true. :)
