  The problem of minimization of the sum of two convex functions has various
theoretical and real-world applications. One of the popular methods for solving
this problem is the proximal gradient method (proximal forward-backward
algorithm). A very common assumption in the use of this method is that the
gradient of the smooth term is globally Lipschitz continuous. However, this
assumption is not always satisfied in practice, thus casting a limitation on
the method. In this paper, we discuss, in a wide class of finite and
infinite-dimensional spaces, a new variant of the proximal gradient method
which does not impose the above-mentioned global Lipschitz continuity
assumption. A key contribution of the method is the dependence of the iterative
steps on a certain telescopic decomposition of the constraint set into subsets.
Moreover, we use a Bregman divergence in the proximal forward-backward
operation. Under certain practical conditions, a non-asymptotic rate of
convergence (that is, in the function values) is established, as well as the
weak convergence of the whole sequence to a minimizer. We also obtain a few
auxiliary results of independent interest.

Optimization and Control;Machine Learning;Functional Analysis;Numerical Analysis