New properties of forward–backward splitting and a practical proximal-descent algorithm

In this paper, we discuss a proximal-descent algorithm for finding a zero of the sum of two maximal monotone operators in a real Hilbert space. Some new properties of forward–backward splitting are given, which extend the well-known properties of the usual projection. Then, they are used to analyze the weak convergence of the proximal-descent algorithm without assuming Lipschitz continuity of the forward operator. We also give a new technique of choosing trial values of the step length involved in an Armijo-like condition, which returns the (not necessarily decreasing) step length self-adaptively. Rudimentary numerical experiments show that it is effective in practical implementations.