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Convergence Analysis of the Relaxed Douglas-Rachford Algorithm

Motivated by nonconvex, inconsistent feasibility problems in imaging, the relaxed alternating averaged reflections algorithm, or relaxed Douglas-Rachford algorithm (DR$λ$), was first proposed over a decade ago. Convergence results for this algorithm are limited either to convex feasibility or consistent nonconvex feasibility with strong assumptions on the regularity of the underlying sets. Using an analytical framework depending only on metric subregularity and pointwise almost averagedness, we analyze the convergence behavior of DR$λ$ for feasibility problems that are both nonconvex and inconsistent. We introduce a new type of regularity of sets, called super-regular at a distance, to establish sufficient conditions for local linear convergence of the corresponding sequence. These results subsume and extend existing results for this algorithm.