Table 1.

All regression methods provided in the flare package. X ∈ ℝ^n×d denotes the design matrix, and y ∈ ℝⁿ denotes the response vector. “L.P.” denotes the general linear programming solver, and “S.O.C.P” denotes the second-order cone programming solver.

Method	Loss function	A	r	Existing solver
LAD Lasso	$L_{\lambda }(\mathit{\alpha })=\frac{1}{n\lambda }{\Vert \mathit{\alpha }\Vert }_1$	X	y	L.P.
SQRT Lasso	$L_{\lambda }(\mathit{\alpha })=\frac{1}{\sqrt{n}\lambda }{\Vert \mathit{\alpha }\Vert }_2$	X	y	S.O.C.P.
ℓq Lasso	$L_{\lambda }(\mathit{\alpha })=\frac{1}{\sqrt[q]{n}\lambda }{\Vert \mathit{\alpha }\Vert }_q$	X	y	None
Dantzig selector	$L_{\lambda }(\mathit{\alpha })=\{\begin{array}{cc}\infty \hfill & \text{if}{\Vert \mathit{\alpha }\Vert }_{\infty }>\lambda \hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$	$\frac{1}{n}{\mathit{X}}^T\mathit{X}$	$\frac{1}{n}{\mathit{X}}^T\mathit{y}$	L.P.