The Krasnosel'skiĭ-Mann Iterative Method: Recent Progress and Applications

Description

This brief explores the Krasnosel'skiĭ-Man (KM) iterative method, which has been extensively employed to find fixed points of nonlinear methods.   Preface Contents 1 Introduction 1.1 Fixed Point Iteration Procedures 1.2 Fixed Point Formulation of Typical Problems 1.3 Outline 2 Notation and Mathematical Foundations 3 The Krasnosel'skiĭ–Mann Iteration 3.1 The Original Krasnosel'skiĭ–Mann Iteration 3.2 Some Results on the Weak and Strong Convergence 3.3 The Krasnosel'skiĭ–Mann Iteration with Perturbations 3.4 The Convergence Rate 4 Relations of the Krasnosel'skiĭ–Mann Iteration and the Operator Splitting Methods 4.1 The Gradient Descent Algorithm 4.2 The Proximal Point Algorithm 4.3 The Operator Splitting Methods 4.3.1 The Forward-Backward Splitting and Backward-Forward Splitting Methods 4.3.2 The Douglas–Rachford Splitting Method 4.3.3 The Davis–Yin Splitting Method 4.3.4 The Primal-Dual Splitting Method 5 The Inertial Krasnosel'skiĭ–Mann Iteration 5.1 General Inertial Krasnosel'skiĭ–Mann Iterations 5.2 The Alternated Inertial Krasnosel'skiĭ–Mann Iteration and the Online Inertial Krasnosel'skiĭ–Mann Iteration 6 The Multi-step Inertial Krasnosel'skiĭ–Mann Iteration 6.1 The Multi-step Inertial Krasnosel'skiĭ–Mann Iteration 6.2 Two Inertial Parameter Sequences That Do Not Depend on the Iterative Sequence 6.3 Some Applications 7 Relaxation Parameters of the Krasnosel'skiĭ–Mann Iteration 7.1 The Approximate Optimal Relaxation Sequence 7.2 Some Variants of the Krasnosel'skiĭ–Mann Iteration Based on the Projection Methods of Variational Inequality Problems 7.3 The Residual Algorithm 8 Two Applications 8.1 The Asynchronous Parallel Coordinate Updates Method 8.2 The Cyclic Coordinate Update Algorithm Conclusion References