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Optimization on Solution Sets of Common Fixed Point Problems

  • Book
  • © 2021

Overview

Authors:
  1. Alexander J. Zaslavski

    1. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel

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  • Studies the influence of computational errors on minimization problems with a convex objective function on a common fixed point set of a finite family of quasi-nonexpansive mappings
  • Highlights the use of Cimmino type subgradient algorithms
  • Highlights the use of the iterative subgradient algorithms
  • Highlights the use of the dynamic string-averaging subgradient algorithm

Part of the book series: Springer Optimization and Its Applications (SOIA, volume 178)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Introduction

    • Alexander J. Zaslavski
    Pages 1-26

  3. Fixed Point Subgradient Algorithm

    • Alexander J. Zaslavski
    Pages 27-102

  4. Proximal Point Subgradient Algorithm

    • Alexander J. Zaslavski
    Pages 103-173

  5. Cimmino Subgradient Projection Algorithm

    • Alexander J. Zaslavski
    Pages 175-216

  6. Iterative Subgradient Projection Algorithm

    • Alexander J. Zaslavski
    Pages 217-242

  7. Dynamic String-Averaging Subgradient Projection Algorithm

    • Alexander J. Zaslavski
    Pages 243-263

  8. Fixed Point Gradient Projection Algorithm

    • Alexander J. Zaslavski
    Pages 265-288

  9. Cimmino Gradient Projection Algorithm

    • Alexander J. Zaslavski
    Pages 289-310

  10. A Class of Nonsmooth Convex Optimization Problems

    • Alexander J. Zaslavski
    Pages 311-397

  11. Zero-Sum Games with Two Players

    • Alexander J. Zaslavski
    Pages 399-425

  12. Back Matter

    Pages 427-434
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Keywords

About this book

This book is devoted to a detailed study of the subgradient projection method and its variants for convex optimization problems over the solution sets of common fixed point problems and convex feasibility problems. These optimization problems are investigated to determine good solutions obtained by different versions of the subgradient projection algorithm in the presence of sufficiently small computational errors.  The use of selected algorithms is highlighted including the Cimmino type subgradient, the iterative subgradient, and the dynamic string-averaging subgradient.  All results presented are new.  Optimization problems where the underlying constraints are the solution sets of other problems, frequently occur in applied mathematics. The reader should not miss the section in Chapter 1 which considers some examples arising in the real world applications. The problems discussed have an important impact in optimization theory as well. The book will be useful for researches interested in the optimization theory and its applications.

Reviews

“Author … make accessible a number of topics that are not often found in many books. … All the algorithms are clearly explained and presented. The results presented in this book will be useful for problems with complicated sets of feasible points arising in engineering, computed tomography and radiation therapy planning. Overall, this book is an excellent contribution to the field of optimization, and it is highly recommended to the students and researchers interested in optimization theory and its applications.” (Samir Kumar Neogy, zbMATH 1479.49001, 2022)

Authors and Affiliations

  • Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel

    Alexander J. Zaslavski

About the author

​Alexander J. Zaslavski is professor in the Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Israel. He has authored numerous books with Springer, the most recent of which include Turnpike Theory for the Robinson–Solow–Srinivasan Model (978-3-030-60306-9),  The Projected Subgradient Algorithm in Convex Optimization (978-3-030-60299-4),  Convex Optimization with Computational Errors (978-3-030-37821-9), Turnpike Conditions in Infinite Dimensional Optimal Control (978-3-030-20177-7).

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