I have a question about the elastic net penalty as implemented in
glmnet in R compared to the original paper by Zou and Hastie (2005). In
glmnet the penalty is listed as
$$ (1-\alpha)/2||\beta||_2^2+α||\beta||_1. $$
but in the paper it is
$$(1-\alpha)||\beta||_1 + \alpha||\beta||_2^2. $$
Does anyone know where the factor $\frac12$ some from? (Never mind the fact that the $\alpha$'s were swapped between the two parameterisations.) In both cases the penalties are multiplied by $\lambda$, but what are the mathematical/technical arguments for not using a simple convex combination of the lasso and ridge penalties?