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It is known that operator splitting methods based on Forward BackwardSplitting (FBS), Douglas-Rachford Splitting (DRS), and Davis-Yin Splitting(DYS) decompose a difficult optimization problems into simpler subproblemsunder proper convexity and smoothness assumptions. In this paper, we identifyan envelope (an objective function) whose gradient descent iteration under avariable metric coincides with DYS iteration. This result generalizes theMoreau envelope for proximal-point iteration and the envelopes for FBS and DRSiterations identified by Patrinos, Stella, and Themelis.  Based on the new envelope and the Stable-Center Manifold Theorem, we furthershow that, when FBS or DRS iterations start from random points, they avoid allstrict saddle points with probability one. This results extends the similarresults by Lee et al. from gradient descent to splitting methods.

An Envelope for Davis-Yin Splitting and Strict Saddle Point Avoidance