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