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

Yanli Liu(yanlimath.ucla.edu)
Wotao Yin(wotaoyinmath.ucla.edu)

Abstract: 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 subproblem 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 result extends the similar results by Lee et al. from gradient descent to splitting methods.

Keywords: Splitting methods, Strict saddle points, Envelope, Stable-Center Manifold Theorem

Category 1: Nonlinear Optimization

Citation: Liu, Yanli, and Wotao Yin. "An Envelope for Davis-Yin Splitting and Strict Saddle Point Avoidance." arXiv preprint arXiv:1804.08739 (2018).

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Entry Submitted: 05/05/2018
Entry Accepted: 05/05/2018
Entry Last Modified: 05/05/2018

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