  We revisit the classical Douglas-Rachford (DR) method for finding a zero of
the sum of two maximal monotone operators. Since the practical performance of
the DR method crucially depends on the stepsizes, we aim at developing an
adaptive stepsize rule. To that end, we take a closer look at a linear case of
the problem and use our findings to develop a stepsize strategy that eliminates
the need for stepsize tuning. We analyze a general non-stationary DR scheme and
prove its convergence for a convergent sequence of stepsizes with summable
increments. This, in turn, proves the convergence of the method with the new
adaptive stepsize rule. We also derive the related non-stationary alternating
direction method of multipliers (ADMM) from such a non-stationary DR method. We
illustrate the efficiency of the proposed methods on several numerical
examples.

Optimization and Control;Numerical Analysis;Machine Learning