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

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  • Yanli Liu

    (University of California, Los Angeles)

  • Wotao Yin

    (University of California, Los Angeles)

Abstract

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.

Suggested Citation

  • Yanli Liu & Wotao Yin, 2019. "An Envelope for Davis–Yin Splitting and Strict Saddle-Point Avoidance," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 567-587, May.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-019-01477-z
    DOI: 10.1007/s10957-019-01477-z
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    References listed on IDEAS

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    1. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    2. Pontus Giselsson & Mattias Fält, 2018. "Envelope Functions: Unifications and Further Properties," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 673-698, September.
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