ASYMPTOTIC CONVERGENCE ANALYSIS OF THE FORWARD-BACKWARD SPLITTING ALGORITHM

The asymptotic convergence of the forward-backward splitting algorithm for solving equations of type 0 is an element of T(z) is analyzed, where T is a multivalued maximal monotone operator in the n-dimensional Euclidean space, When the problem has a nonempty solution set, and T is split in the form T = J + h It with J being maximal monotone and h being co-coercive with modulus greater than 1/2, convergence rates are shown, under mild conditions, to be linear, superlinear or sublinear depending on how rapidly J(-1) and h(-1) grow in the neighborhoods of certain specific points. As a special case, when both J and h are polyhedral functions, we get R-linear convergence and 2-step e-linear convergence without any further assumptions on the strict monotonicity on T or on the uniqueness of the solution. As another special case when h = 0, the splitting algorithm reduces to the proximal point algorithm, and we get new results, which complement R. T. Rockafellar's and F. J. Luque's earlier results on the proximal point algorithm.