Relatively Inexact Proximal Point Algorithm and Linear Convergence Analysis

Based on a notion of relatively maximal m -relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar’s theorem 1976 on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this newmodel, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.