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We analyze Douglas-Rachford splitting techniques applied to solving weaklyconvex optimization problems. Under mild regularity assumptions, and by thetoken of a suitable merit function, we show convergence to critical points andlocal linear rates of convergence. The merit function, comparable to the Moreauenvelope in Variational Analysis, generates a descent sequence, a feature thatallows us to extend to the non-convex setting arguments employed in convexoptimization. A by-product of our approach is a ADMM-like method forconstrained problems with weakly convex objective functions. When specializedto multistage stochastic programming, the proposal yields a nonconvex versionof the Progressive Hedging algorithm that converges with linear speed. Thenumerical assessment on a battery of phase retrieval problems shows promisingnumerical performance of our method, when compared to existing algorithms inthe literature.

Convergence Rate of Nonconvex Douglas-Rachford splitting via merit functions, with applications to weakly convex constrained optimization