  We consider regularized least-squares problems of the form $\min_{x}
\frac{1}{2}\Vert Ax - b\Vert_2^2 + \mathcal{R}(Lx)$. Recently, Zheng et al.,
2019, proposed an algorithm called Sparse Relaxed Regularized Regression (SR3)
that employs a splitting strategy by introducing an auxiliary variable $y$ and
solves $\min_{x,y} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \frac{\kappa}{2}\Vert Lx
- y\Vert_2^2 + \mathcal{R}(x)$. By minimizing out the variable $x$ we obtain an
equivalent system $\min_{y} \frac{1}{2} \Vert F_{\kappa}y -
g_{\kappa}\Vert_2^2+\mathcal{R}(y)$. In our work we view the SR3 method as a
way to approximately solve the regularized problem. We analyze the conditioning
of the relaxed problem in general and give an expression for the SVD of
$F_{\kappa}$ as a function of $\kappa$.
  Furthermore, we relate the Pareto curve of the original problem to the
relaxed problem and we quantify the error incurred by relaxation in terms of
$\kappa$. Finally, we propose an efficient iterative method for solving the
relaxed problem with inexact inner iterations. Numerical examples illustrate
the approach.
