{k=1}^K \sum_{i=1}^n  \left[\frac{1}{2}\left(y_i-c_k\sum_{j\in C_k} X_{ij}\beta_j \right)^2+\lambda\left(\sum_{j \in C_k} \alpha|\beta_j| + 
    \frac{(1-\alpha)}{2}||\beta_j||_2^2 \right) \; \right]
\label{eqn:obj}\end{aligned}$$ subject to $c_k\geq 0 \; \forall k$. Our algorithm (detailed below) sets $c_k=1 \; \forall k$, optimizes over $\beta$, and then optimizes over $c$.

Consider an extreme case where the sample covariance matrix happens to be block diagonal with $K$ connected components. This occurs when predictors in different correlated groups are orthogonal to each other. The subgradient equation of the lasso splits naturally into separate systems of equations as in equation (\[eqn:subg2\]) for the component lasso. The lasso coefficients will be identical to the component lasso coefficients before the NNLS step, which reweights the predictors corresponding to each component.

This can be easily extended to the elastic net. Let $\lambda'=\frac{\lambda(1-\alpha)}{2}$ be the tuning parameter corresponding to the $\ell_2$ penalty. The naive elastic net problem can be written as a lasso problem on an augmented data set $(y^*,X^*)$, where $y^*=(y,0)$ is now an $n+p$ vector and $$X^*=(1+\lambda')^{-1/2}\left[
\begin{array}{c}
X \\
\sqrt {\lambda'} I
\end{array}\right]
.$$ The sample covariance matrix corresponding to the augmented observations $X^*$ $$(1+\lambda')^{-1}(X^TX + \lambda'I)$$ is clearly block diagonal when the predictors in different components are orthogonal.

Therefore, the subgradient equation of the 