Reviews: Generalized Linear Model Regression under Distance-to-set Penalties

I found much to like about this paper. It was very clear and well-written, and the proposed method is elegant, intuitive, novel (to the best of my knowledge), seemingly well motivated, and widely applicable. In addition, the main ideas in the paper are well-supported by appropriate numerical and real data examples. I would like to see the authors compare their method to two other methods that I would see as major competitors: first, the relaxed LASSO (Meinshausen 2007) wherein we fit the lasso and then do an unpenalized fit using only the variables that the lasso has selected. As I understand the field, this is the most popular method and the one that Hastie, Tibshirani, and Wainwright (2015) recommend for users who want to avoid the bias of the lasso (but note that the shrinkage "bias" is sometimes desirable to counteract the selection bias or "winner's curse" that the selected coefficients may suffer from). Second, best-subsets itself is now feasible using modern mixed-integer optimization methods (Bertsimas, Kind, and Mazumder, 2016) (if one of these competitors outperforms your method, it would not make me reconsider my recommendation to accept, since the method you propose applies to a much more general class of problems).