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.
In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas–Rachford splitting. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas–Rachford splitting diverge. Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility. As a first-order method, the proposed algorithm relies on simple subroutines, and therefore is simple to implement and has low per-iteration cost.