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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?

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Apparently, the problem has been solved recently: Peng Zeng, Yu Zhu, and Tianhong He, Linearly Constrained Lasso with Application in Glioblastoma Data. I haven't read it yet, but it looks like they have designed a LARS type algorithm. – Taha Jul 31 '12 at 23:26
Also any Lasso algorithm with projecting to constraint set after each iteration also should work. – mirror2image Aug 1 '12 at 7:42

"The Constrained LASSO" by Gareth M James, Courtney Paulson, and Paat Rusmevichientong which is under review and linked below

seems to have solved this specific problem.

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The enclosed paper developed a constrained LASSO for portfolio optimisation:

Sparse and stable Markowitz portfolios (2009). Proceedings of the National Academy of Science, Vol. 106, No. 30, pages 12267-12272.

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