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The problem of minimization of the sum of two convex functions has varioustheoretical and real-world applications. One of the popular methods for solvingthis problem is the proximal gradient method (proximal forward-backwardalgorithm). A very common assumption in the use of this method is that thegradient of the smooth term is globally Lipschitz continuous. However, thisassumption is not always satisfied in practice, thus casting a limitation onthe method. In this paper, we discuss, in a wide class of finite andinfinite-dimensional spaces, a new variant of the proximal gradient methodwhich does not impose the above-mentioned global Lipschitz continuityassumption. A key contribution of the method is the dependence of the iterativesteps on a certain telescopic decomposition of the constraint set into subsets.Moreover, we use a Bregman divergence in the proximal forward-backwardoperation. Under certain practical conditions, a non-asymptotic rate ofconvergence (that is, in the function values) is established, as well as theweak convergence of the whole sequence to a minimizer. We also obtain a fewauxiliary results of independent interest.

A telescoping Bregmanian proximal gradient method without the global Lipschitz continuity assumption