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We develop a new variational approach on level sets aiming towardsconvergence rate analysis of a variable Bregman proximal gradient (VBPG) methodfor a broad class of nonsmooth and nonconvex optimization problems. With thisnew approach, we are able to extend the concepts of Bregman proximal mappingand their corresponding Bregman proximal envelops, Bregman proximal gapfunction to nonconvex setting. Properties of these mappings and functions areexamined. An aim of this work is to provide a solid foundation on which furtherdesign and analysis of VBPG for more general nonconvex optimization problemsare possible. Another aim is to provide a unified theory on linear convergenceof VBPG with a particular interest towards proximal gradient methods. Centrolto our analysis for achieving the above goals is an error bound in terms oflevel sets and subdifferentials (level-set subdifferential error bound) alongwith its links to other level-set error bounds. As a consequence, we haveestablished a number of positive results. These newly established results notonly enable us to show that any accumulation of the sequence generated by VBPGis at least a critical point of the limiting subdifferential or even a criticalpoint of the proximal subdifferential with a fixed Bregman function in eachiteration, but also provide a fresh perspective that allows us to exploreinner-connections among many known sufficient conditions for linear convergenceof various first-order methods. Along the way, we are able to derive a numberof verifiable conditions for level-set error bounds to hold, obtain linearconvergence of VBPG, and derive necessary conditions and sufficient conditionsfor linear convergence relative to a level set for nonsmooth and nonconvexoptimization problems.

A Variational Approach on Level sets and Linear Convergence of Variable Bregman Proximal Gradient Method for Nonconvex Optimization Problems