  We introduce Bella, a locally superlinearly convergent Bregman forward
backward splitting method for minimizing the sum of two nonconvex functions,
one of which satisfying a relative smoothness condition and the other one
possibly nonsmooth. A key tool of our methodology is the Bregman
forward-backward envelope (BFBE), an exact and continuous penalty function with
favorable first- and second-order properties, and enjoying a nonlinear error
bound when the objective function satisfies a Lojasiewicz-type property. The
proposed algorithm is of linesearch type over the BFBE along candidate update
directions, and converges subsequentially to stationary points, globally under
a KL condition, and owing to the given nonlinear error bound can attain
superlinear convergence rates even when the limit point is a nonisolated
minimum, provided the directions are suitably selected.
