Super-Relaxed ()-Proximal Point Algorithms, Relaxed ()-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions

Ravi P Agarwal,Ram U Verma

doi:10.1155/2009/957407

Abstract

We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed ( )-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal ( )-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the relaxed proximal point algorithm by Eckstein and Bertsekas (1992). Even for the linear convergence analysis for the overrelaxed (or super-relaxed) ( )-proximal point algorithm, the fundamental model for Rockafellar's case does the job. Furthermore, we attempt to explore possibilities of generalizing the Yosida regularization/approximation in light of maximal ( )-monotonicity, and then applying to first-order evolution equations/inclusions.

Highlights

Introduction and PreliminariesWe begin with a real Hilbert space X with the norm · and the inner product ·, ·
We focus mostly on applications of the super-relaxed η proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal η -monotonicity
1.2 converges weakly to a solution of 1.1, provided that the approximation is made sufficiently accurate as the iteration proceeds, where Pk I ckM −1 for a sequence {ck} of positive real numbers that is bounded away from zero, and in second part using the first part and further amending the proximal point algorithm succeeded in achieving the linear convergence

Summary

Introduction and Preliminaries

We begin with a real Hilbert space X with the norm · and the inner product ·, ·. As a matter of fact, Eckstein and Bertsekas 2 applied Algorithm 1.4 to approximate a weak solution to 1.1 In other words, they established Theorem 1.1 using the relaxed proximal point algorithm instead. Verma 3 generalized the relaxed proximal point algorithm and applied to the approximation solvability of variational inclusion problems of the form 1.1. We start with some introductory materials to the over-relaxed η proximal point algorithm based on the notion of maximal η -monotonicity, and recall some investigations on approximation solvability of a general class of nonlinear inclusion problems involving maximal η -monotone mappings in a Hilbert space setting.

General Maximal η-Monotonicity

Some Specializations

36 We further examine the estimate