Level-Set Subdifferential Error Bounds and Linear Convergence of Bregman Proximal Gradient Method

Daoli Zhu (), Sien Deng (), Minghua Li () and Lei Zhao ()
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Daoli Zhu: Shanghai Jiao Tong University
Sien Deng: Northern Illinois University
Minghua Li: Chongqing University of Arts and Sciences
Lei Zhao: Shanghai Jiao Tong University

Journal of Optimization Theory and Applications, 2021, vol. 189, issue 3, No 10, 889-918

Abstract: Abstract In this work, we develop a level-set subdifferential error bound condition with an eye toward convergence rate analysis of a variable Bregman proximal gradient (VBPG) method for a broad class of nonsmooth and nonconvex optimization problems. It is proved that the aforementioned condition guarantees linear convergence of VBPG and is weaker than Kurdyka–Łojasiewicz property, weak metric subregularity, and Bregman proximal error bound. Along the way, we are able to derive a number of verifiable conditions for level-set subdifferential error bounds to hold, and necessary conditions and sufficient conditions for linear convergence relative to a level set for nonsmooth and nonconvex optimization problems. The newly established results not only enable us to show that any accumulation point of the sequence generated by VBPG is at least a critical point of the limiting subdifferential or even a critical point of the proximal subdifferential with a fixed Bregman function in each iteration, but also provide a fresh perspective that allows us to explore inner-connections among many known sufficient conditions for linear convergence of various first-order methods.

Keywords: Level-set subdifferential error bound; Variable Bregman proximal gradient method; Linear convergence; Bregman proximal error bound (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-021-01865-4

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