This is a short follow up on my last post where I wrote about the sweet spot of the stepsize of the Douglas-Rachford iteration. For the case \beta-Lipschitz + \mu-strongly monotone, the iteration with stepsize t converges linear with rate

\displaystyle r(t) = \tfrac{1}{2(1+t\mu)}\left(\sqrt{2t^{2}\mu^{2}+2t\mu + 1 +2(1 - \tfrac{1}{(1+t\beta)^{2}} - \tfrac1{1+t^{2}\beta^{2}})t\mu(1+t\mu)} + 1\right)

Here is animated plot of this contraction factor depending on \beta and \mu and t acts as time variable:


What is interesting is, that this factor has increasing or decreasing in t depending on the values of \beta and \mu.

For each pair (\beta,\mu) there is a best t^* and also a smallest contraction factor r(t^*). Here are plots of these quantities:

Comparing the plot of te optimal contraction factor to the animated plot above, you see that the right choice of the stepsize matters a lot.