Hybrid inexact proximal point algorithms based on RMM frameworks with applications to variational inclusion problems

• Published in: Journal of applied mathematics & computing Vol. 39; no. 1-2; pp. 345 - 365
• Main Author: Verma, Ram U.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.06.2012; Springer Nature B.V
• Subjects: Algorithms; Applied Mathematics; Approximation; Computational Mathematics and Numerical Analysis; Convergence; Evolution; Hilbert space; Inclusions; Inverse; Mapping; Mathematical analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Operators; Studies; Theory of Computation; Maximal monotone mapping; 47H10; Relatively; Generalized resolvent operator; 49J40; Inclusion problems
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-011-0529-5

A new application-oriented notion of relatively A-maximal monotonicity (RMM) framework is introduced, and then it is applied to the approximation solvability of a general class of inclusion problems, while generalizing other existing results on linear convergence, including Rockafellar’s theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. The obtained results not only generalize most of the existing investigations, but also reduce smoothly to the case of the results on maximal monotone mappings and corresponding classical resolvent operators. Furthermore, our proof approach differs significantly to that of Rockafellar’s celebrated work, where the Lipschitz continuity of M −1 , the inverse of M : X →2 X , at zero is assumed to achieve a linear convergence of the proximal point algorithm. Note that the notion of relatively A -maximal monotonicity framework seems to be used to generalize the classical Yosida approximation (which is applied and studied mostly based on the classical resolvent operator in the literature) that in turn can be applied to first-order evolution equations as well as evolution inclusions.