Can I penalize an arbitrary regression model and get Elastic-Net-esque results? Consider an arbitrary-ish regression model with the unpenalized likelihood
$$
\log \mathcal{L} = \sum_i f\left(y_i\,|\,g(\beta_0 + \beta x_i)\right)
$$
with $\beta = \left(\beta_1, \dots, \beta_K\right)$ and $x_i$ being a data vector.
If I instead maximize
$$
\log \mathcal{L} = \sum_i f\left(y_i\,|\,g(\beta_0 + \beta x_i)\right) + \lambda P_\alpha(\beta)
$$
where $P_\alpha(\beta) = \sum_{k=1}^K \frac{1}{2}(1-\alpha)\beta_k^2 + \alpha | \beta_k |$, will my estimated $\beta$ have the same nice properties for an arbitrary convex $f$ as for a well-studied $f$ like the Bernoulli or Gaussian likelihoods studied in Friedman, Hastie, and Tibshirani (2010)?
Specifically, what if $y$ is censored? That is, $y = \cases{y^* &\text{if}\ y^* < c \\ c &\text{if}\ y^* \geq c}$. Then my likelihood is something like
$$
\log \mathcal{L} = \sum_{i\ : y^*_i < c} F^*\left(y^*_i\,|\,g(\beta_0 + \beta x_i)\right)\,f^*\left(y^*_i\,|\,g(\beta_0 + \beta x_i)\right)
\\+ \sum_{i\ : y^*_i \geq c} 1 - F^*(c\,|\,g(\beta_0 + \beta x_i))
\\+ \lambda P_\alpha(\beta)
$$
Is there any reason why this wouldn't work?
I imagine this is related to the question Do I get the nice asymptotic properties of MLE when I restrict the parameter space?
 A: I'll try to address this question in a general way.  For generalized linear models (GLM), it makes complete sense to use elastic-net priors on the parameters as you describe.  This is merely to say that the standard elastic-net regularization framework works out-of-the-box for a broader class of models than simply Bernoulli and Gaussian observation models which are simple examples of GLMs, (e.g. Poisson observations correspond to another GLM).
Each GLM has a canonical link and when using this link function, the GLM is log-concave, and the negative log likelihood (NLL) is convex.  The L1 and L2 penalties are convex.  The sum of these convex penalties with the convex objective (NLL) gives an overall convex objective. 
The possible issues arise is if you use a non-standard link (specific non-standard links won't break things) or aren't using a GLM.  Basically for an arbitrary regression objective, it may be the case that the NLL part of your objective will no longer be convex (although the penalties will remain convex). Even if the NLL is not convex, the penalties will affect the estimated weights, but guarantees are weaker since you will obtain locally optimal weights.   
For specific examples you have, you could check whether your objective function is convex before the addition of penalties.  If it is, then the addition of convex penalties usually preserves convexity.  If the original objective is not convex, then the messiness is probably coming from that rather than the penalties. 
