International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 691952, 11 pages
doi:10.1155/2009/691952

Relatively inexact proximal point algorithm and linear convergence analysis

Abstract

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 new model, 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.