A note on the Douglas–Rachford splitting method for optimization problems involving hypoconvex functions

Recently, the convergence of the Douglas–Rachford splitting method (DRSM) was established for minimizing the sum of a nonsmooth strongly convex function and a nonsmooth hypoconvex function under the assumption that the strong convexity constant $$\beta $$ is larger than the hypoconvexity constant $$\omega $$ . Such an assumption, implying the strong convexity of the objective function, precludes many interesting applications. In this paper, we prove the convergence of the DRSM for the case $$\beta =\omega $$ , under relatively mild assumptions compared with some existing work in the literature.