Numerical Optimization with Computational Errors

✍ Scribed by Alexander J. Zaslavski (auth.)

• Publisher: Springer International Publishing
• Year: 2016
• Tongue: English
• Leaves: 308
• Series: Springer Optimization and Its Applications 108
• Edition: 1
• Category: Library

✦ Synopsis

This book studies the approximate solutions of optimization problems in the presence of computational errors. A number of results are presented on the convergence behavior of algorithms in a Hilbert space; these algorithms are examined taking into account computational errors. The author illustrates that algorithms generate a good approximate solution, if computational errors are bounded from above by a small positive constant. Known computational errors are examined with the aim of determining an approximate solution. Researchers and students interested in the optimization theory and its applications will find this book instructive and informative.

This monograph contains 16 chapters; including a chapters devoted to the subgradient projection algorithm, the mirror descent algorithm, gradient projection algorithm, the Weiszfelds method, constrained convex minimization problems, the convergence of a proximal point method in a Hilbert space, the continuous subgradient method, penalty methods and Newton’s method.

✦ Table of Contents

Front Matter....Pages i-ix
Introduction....Pages 1-9
Subgradient Projection Algorithm....Pages 11-40
The Mirror Descent Algorithm....Pages 41-58
Gradient Algorithm with a Smooth Objective Function....Pages 59-72
An Extension of the Gradient Algorithm....Pages 73-84
Weiszfeld’s Method....Pages 85-103
The Extragradient Method for Convex Optimization....Pages 105-118
A Projected Subgradient Method for Nonsmooth Problems....Pages 119-136
Proximal Point Method in Hilbert Spaces....Pages 137-147
Proximal Point Methods in Metric Spaces....Pages 149-168
Maximal Monotone Operators and the Proximal Point Algorithm....Pages 169-181
The Extragradient Method for Solving Variational Inequalities....Pages 183-203
A Common Solution of a Family of Variational Inequalities....Pages 205-224
Continuous Subgradient Method....Pages 225-238
Penalty Methods....Pages 239-264
Newton’s Method....Pages 265-296
Back Matter....Pages 297-304

✦ Subjects

Calculus of Variations and Optimal Control; Optimization; Numerical Analysis; Operations Research, Management Science