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

It is known that operator splitting methods based on forward–backward splitting, Douglas–Rachford splitting, and Davis–Yin splitting decompose 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 Davis–Yin splitting iteration. This result generalizes the Moreau envelope for proximal-point iteration and the envelopes for forward–backward splitting and Douglas–Rachford splitting iterations identified by Patrinos, Stella, and Themelis. Based on the new envelope and the stable–center manifold theorem, we further show that, when forward–backward splitting or Douglas–Rachford splitting iterations start from random points, they avoid all strict saddle points with probability one. This result extends the similar results by Lee et al. from gradient descent to splitting methods.