TITLE: What Numerical Methods Are Known to Solve $ {L}_{1} $ Regularized Quadratic Programming Problems?
QUESTION [9 upvotes]: What numerical methods are suitable to solve the following problem
$$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$
where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive definite, and $\lambda\in\mathbf{R}$ is positive?
Typically this problem arises in L1-regularized regression problems (e.g. LASSO) but the common method to solve that (LARS) requires that you can compute regression residuals, which you can't in this case.
Are there standard tools (open source or commercial) that can solve problems like this?

REPLY [3 votes]: CVX (MATLAB) or CVXPY (Python) would allow you to solve that really easily, as long as your problem isn't too large.
The CVX code would be just:
cvx_begin
    variable x(n)
    minimize( 1/2*quad_form(x,A) + b'*x + lambda*norm(x,1) )
cvx_end