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

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 η proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal η -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 η -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 η -monotonicity, and then applying to firstorder evolution equations/inclusions.