Abstract
The asymptotic convergence of the forward-backward splitting algorithm for solving equations of type 0 ∈ 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 = ℱ + h with ℱ being maximal monotone and h being co-coercive with modulus greater than ½, convergence rates are shown, under mild conditions, to be linear, superlinear or sublinear depending on how rapidly ℱ−1 and h−1 grow in the neighborhoods of certain specific points. As a special case, when both ℱ and h are polyhedral functions, we get R-linear convergence and 2-step Q-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.
Published Version
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