Elastic Net Regularization

Definition

$\begin{array}{rl}{\hat{\beta }}_{\text{lasso}}& =\underset{\beta }{\mathrm{argmin}}(\mathbf{y}-\mathbf{X}\beta {)}^{⊺}(\mathbf{y}-\mathbf{X}\beta )+\lambda (\alpha {\beta }^{⊺}\beta +(1-\alpha )||\beta ||)\\ & =\underset{\beta }{\mathrm{argmin}}(\mathbf{y}-\mathbf{X}\beta {)}^{⊺}(\mathbf{y}-\mathbf{X}\beta ),\text{subject to}(\alpha {\beta }^{⊺}\beta +(1-\alpha )||\beta ||)\le c\end{array}$

Elastic net is a regularized regression method that linearly combined the penalties of the lasso and ridge regressions.