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

  • Yanli LiuEmail author
  • Wotao Yin


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.


Splitting methods Strict saddle points Envelope Stable–center manifold theorem 

Mathematics Subject Classification

37L10 49J52 65K05 65K10 90C26 



This work is supported in part by NSF Grant DMS-1720237 and ONR Grant N000141712162.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of California, Los AngelesLos AngelesUSA

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