Least Squares with $ {L}_{1} $ Regularization (Lasso) and Linear Inequality Constraints

Question

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?

Answer 1

"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.