  Lipschitz continuity of the gradient mapping of a continuously differentiable
function plays a crucial role in designing various optimization algorithms.
However, many functions arising in practical applications such as low rank
matrix factorization or deep neural network problems do not have a Lipschitz
continuous gradient. This led to the development of a generalized notion known
as the $L$-smad property, which is based on generalized proximity measures
called Bregman distances. However, the $L$-smad property cannot handle
nonsmooth functions, for example, simple nonsmooth functions like $\abs{x^4-1}$
and also many practical composite problems are out of scope. We fix this issue
by proposing the MAP property, which generalizes the $L$-smad property and is
also valid for a large class of nonconvex nonsmooth composite problems. Based
on the proposed MAP property, we propose a globally convergent algorithm called
Model BPG, that unifies several existing algorithms. The convergence analysis
is based on a new Lyapunov function. We also numerically illustrate the
superior performance of Model BPG on standard phase retrieval problems, robust
phase retrieval problems, and Poisson linear inverse problems, when compared to
a state of the art optimization method that is valid for generic nonconvex
nonsmooth optimization problems.
