I want to efficiently solve the following optimization problem:
\begin{align}
\min &\quad \left\|\mathbf{x}-\mathbf{x}_0\right\|_2^2 + \lambda\left\|\mathbf{x}\right\|_1\\
\text{Subject to}& \quad A\mathbf{x} \leq c,
\end{align}
where $\mathbf{x}\in \mathbb{R}^{n\times 1}$, for big values of $n$. I tried coordinate descend, but it doesn't work. I also don't want to formulate it as QP and use interior-point methods because they are slow. Any idea?
Best Answer
"The Constrained LASSO" by Gareth M James, Courtney Paulson, and Paat Rusmevichientong which is under review and linked below
http://www-bcf.usc.edu/~gareth/research/CLassoFinal.pdf
seems to have solved this specific problem.