variables and the usual linear regression setup: $$\label{eq:model}
y = X\beta^* + \sigma\epsilon,$$ where $\beta^* \in \R^p$ are unknown coefficients to be estimated, $\sigma^2>0$ is the noise variance, and the components of the noise vector $\epsilon \in \R^n$ are i.i.d. with $\E[\epsilon_i]=0$ and $\Var(\epsilon_i)=1$. We assume that $y$ has been centered, and the columns of $X$ are centered and scaled, so that we can omit an intercept in the model. The lasso estimator [@lasso; @bp], is defined as $$\label{eqn:lasso}
\hbeta = {\mathop{\mathrm{argmin}}}_{\beta\in\R^p} \, 
\half\|y-X\beta\|_2^2 + \lambda\|\beta\|_1,$$ where $\lambda \geq 0$ is a tuning parameter, controlling the degree of sparsity in the estimate $\hbeta$.

Variable selection is important in many modern applications, for which the lasso has proven to be successful. However, this method has known limitations in certain settings: there is a solution with at most $n$ non-zero coefficients when $p>n$, and if a group of relevant variables is highly correlated, it tends to include only one in the model. [These conditions occur frequently in real applications, such as genomics, where we often have a large number of predictors that can be divided into highly correlated groups. It is therefore of practical interest to overcome these limitations.]{}

The elastic net  