**Contemporary Mathematics**

Volume: 636;
2015;
266 pp;
Softcover

MSC: Primary 15; 40; 41; 46; 47; 49; 54; 58; 62; 65; 90;

**Print ISBN: 978-1-4704-1480-1
Product Code: CONM/636**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-2275-2
Product Code: CONM/636.E**

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# Infinite Products of Operators and Their Applications

Share this page *Edited by *
*Simeon Reich; Alexander J. Zaslavski*

A co-publication of the AMS and Bar-Ilan University

This volume contains the proceedings of the workshop on Infinite Products of
Operators and Their Applications, held from May 21–24, 2012, at the
Technion-Israel Institute of Technology, Haifa, Israel.

The papers cover many different topics regarding infinite products
of operators and their applications: projection methods for solving
feasibility and best approximation problems, arbitrarily slow
convergence of sequences of linear operators, monotone operators,
proximal point algorithms for finding zeros of maximal monotone
operators in the presence of computational errors, the
Pascoletti-Serafini problem, remetrization for infinite families of
mappings, Poisson's equation for mean ergodic operators, vector-valued
metrics in fixed point theory, contractivity of infinite products and
mean convergence theorems for generalized nonspreading mappings.

#### Readership

Graduate students and research mathematicians interested in applied functional analysis, operator theory, and optimization.

# Table of Contents

## Infinite Products of Operators and Their Applications

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Conference Program ix10 free
- Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces 114 free
- Projectors on Intersections of Subspaces 4154
- Recent Progress on Monotone Operator Theory 5164
- Feasibility-Seeking and Superiorization Algorithms Applied to Inverse Treatment Planning in Radiation Therapy 8396
- Arbitrarily Slow Convergence of Sequences of Linear Operators 93106
- 1. Introduction 93106
- 2. Types of Convergence 95108
- 3. A Characterization of Almost Arbitrarily Slow Convergence 98111
- 4. Arbitrarily Slow Convergence: A Useful Sufficient Condition 100113
- 5. Trichotomy for Powers of an Operator 100113
- 6. The Bernstein and Tyuriemskiih Lethargy Theorems 103116
- 7. Application to Positive Linear Operators 106119
- 8. Application to Quadrature Rules 109122
- 9. Application to Cyclic Projections 110123
- 10. Application to Intermittent Projections 113126
- 11. Application to a Theorem of Xu-Zikatanov 115128
- 12. Further Applications 116129
- References 117130

- Reformulating the Pascoletti-Serafini Problem as a Bi-Level Optimization Problem 121134
- Remetrization Theorems for Finite Families of Mappings and Hyperbolic Iterated Function Systems 131144
- Poisson’s Equation for Mean Ergodic Operators 141154
- Vector-Valued Metrics in Fixed Point Theory 149162
- Projected Gradient Method for Non-Negative Least Square 167180
- 1. Introduction 167180
- 2. Problem formulation and some preliminary results 168181
- 3. Projected gradient method 169182
- 4. Fast Projected Gradient 171184
- 5. Projected Gradient for full rank NNLS 173186
- 6. Projected Gradient Method for SVM 175188
- 7. Concluding Remarks 176189
- 8. Appendix 1 176189
- 9. Appendix 2 177190
- Acknowledgement 178191
- References 178191

- Nonlinear Equilibrium for Resource Allocation Problems 181194
- 1. Introduction 181194
- 2. Basic Assumptions. 183196
- 3. Generalized Walras-Wald Equilibrium. 184197
- 4. Pseudo-Gradient Projection Method 186199
- 5. Extra Pseudo-Gradient Method for finding NE 189202
- 6. Convergence rate of the EPG method 192205
- 7. Appendix 194207
- 8. Concluding Remarks 195208
- References 196209

- Infinite Products of Discontinuous Operators 199212
- Contractivity, Porosity and Infinite Products 203216
- Performance of Hull-Detection Algorithms for Proton Computed Tomography Reconstruction 211224
- Attractive Point and Mean Convergence Theorems for New Generalized Nonspreading Mappings in Banach Spaces 225238
- Proximal Point Algorithms for Finding Zeros of Maximal Monotone Operators in the Presence of Computational Errors 249262
- 1. Introduction 249262
- 2. The proximal point algorithm in a finite-dimensional space 250263
- 3. The proximal point algorithm for finding a common zero of a finite family of maximal monotone operators 253266
- 4. Exact iterates of the proximal point algorithm 258271
- 5. An example 259272
- 6. Inexact iterates of the proximal point algorithm in an infinite-dimensional space 260273
- References 264277

- Back Cover Back Cover1282