Bregman Forward-Backward Operator Splitting

We propose an iterative method for finding a zero of the sum of two maximally monotone operators in reflexive Banach spaces. One of the operators is single-valued, and the method alternates an explicit step on this operator and an implicit step on the other one. Both steps involve the gradient of a convex function that is free to vary over the iterations. The convergence of the resulting forward-backward splitting method is analyzed using the theory of Legendre functions, under a novel assumption on the single-valued operator that captures various existing properties. When applied to minimization problems, rates are obtained for the objective values. The proposed framework unifies and extends several iterative methods which have thus far not been brought together, and it is also new in Euclidean spaces.