  In this work, we develop a level-set subdifferential error bound condition
aiming towards 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-Lojasiewicz
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 acritical 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.
