# Relaxation

The (unscaled, see Non-identity DR Scaling) SCS update equations are:

\[\begin{split}\begin{align} \tilde u^{k+1} &= (I + \mathcal{Q})^{-1} w^k \\ u^{k+1} &= (I + N_{\mathcal{C}_+})^{-1} (2 \tilde u^{k+1} - w^k) \\ w^{k+1} &= w^k + \alpha (u^{k+1} - \tilde u^{k+1}) \\ \end{align}\end{split}\]

where \(\alpha \in (0,2)\) is the relaxation parameter. Vanilla Douglas-Rachford corresponds to setting \(\alpha = 1\). If \(\alpha < 1\) it is referred to as under-relaxation, if \(\alpha > 1\) it is over-relaxation. Typically values of \(\alpha \approx 1.5\) work well. It is controlled by the `alpha` setting. If \(\alpha = 2\) the method reduces to Peaceman-Rachford splitting, which is not guaranteed to converge in general (but will if, for example, the operators in the problem are both maximal strongly monotone).

Thankfully, there is no interaction between \(\alpha\) and the scaling described in Non-identity DR Scaling, and they can be combined immediately.