**Also available as eBook on:**

**Iterative Optimization in Inverse Problems** brings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author’s considerable research in the field, including his recently developed class of SUMMA algorithms. Related to sequential unconstrained minimization methods, the SUMMA class includes a wide range of iterative algorithms well known to researchers in various areas, such as statistics and image processing.

Organizing the topics from general to more specific, the book first gives an overview of sequential optimization, the subclasses of auxiliary-function methods, and the SUMMA algorithms. The next three chapters present particular examples in more detail, including barrier- and penalty-function methods, proximal minimization, and forward-backward splitting. The author also focuses on fixed-point algorithms for operators on Euclidean space and then extends the discussion to include distance measures other than the usual Euclidean distance. In the final chapters, specific problems illustrate the use of iterative methods previously discussed. Most chapters contain exercises that introduce new ideas and make the book suitable for self-study.

Unifying a variety of seemingly disparate algorithms, the book shows how to derive new properties of algorithms by comparing known properties of other algorithms. This unifying approach also helps researchers—from statisticians working on parameter estimation to image scientists processing scanning data to mathematicians involved in theoretical and applied optimization—discover useful related algorithms in areas outside of their expertise.

**Background **Overview

An Urns Model for Remote Sensing

Hidden Markov Models

Measuring the Fourier Transform

Transmission Tomography

Emission Tomography

A Unifying Framework

**Sequential Optimization **Overview

Examples of SUM

Auxiliary-Function Methods

The SUMMA Class of AF Methods

**Barrier-Function and Penalty-Function Methods **Barrier Functions

Examples of Barrier Functions

Penalty Functions

Examples of Penalty Functions

Basic Facts

**Proximal Minimization **The Basic Problem

Proximal Minimization Algorithms

Some Obstacles

All PMA Are SUMMA

Convergence of the PMA

The Non-Differentiable Case

The IPA

Projected Gradient Descent

Relaxed Gradient Descent

Regularized Gradient Descent

The Projected Landweber Algorithm

The Simultaneous MART

A Convergence Theorem

Another Job for the PMA

The Goldstein-Osher Algorithm

A Question

**The Forward-Backward Splitting Algorithm**Moreau’s Proximity Operators

The FBS Algorithm

Convergence of the FBS Algorithm

Some Examples

Minimizing

*f*

_{2 }over a Linear Manifold

Feasible-Point Algorithms

**Operators **Overview

Operators

Contraction Operators

Convex Sets in R

^{J }Orthogonal Projection Operators

Firmly Nonexpansive Gradients

Exercises

**Averaged and Paracontractive Operators **Averaged Operators

Gradient Operators

Two Useful Identities

The Krasnosel’skii-Mann-Opial Theorem

Affine Linear Operators

Paracontractive Operators

Exercises

**Convex Feasibility and Related Problems **Convex Constraint Sets

Using Orthogonal Projections

The ART

Regularization

Avoiding the Limit Cycle

Exercises

**Eigenvalue Bounds **Introduction and Notation

Overview

Cimmino’s Algorithm

The Landweber Algorithms

Some Upper Bounds for

*L*

Simultaneous Iterative Algorithms

Block-Iterative Algorithms

Exercises

**Jacobi and Gauss-Seidel Methods **The Jacobi and Gauss-Seidel Methods: An Example

Splitting Methods

Some Examples of Splitting Methods

Jacobi’s Algorithm and JOR

The Gauss-Seidel Algorithm and SOR

Summary

**The SMART and EMML Algorithms **The SMART Iteration

The EMML Iteration

The EMML and the SMART as AM

The SMART as SUMMA

The SMART as PMA

Using KL Projections

The MART and EMART Algorithms

Extensions of MART and EMART

Convergence of the SMART and EMML

Regularization

Modifying the KL Distance

The ABMART Algorithm

The ABEMML Algorithm

**Alternating Minimization **Alternating Minimization

Exercises

**The EM Algorithm **Overview

A Non-Stochastic Formulation of EM

The Stochastic EM Algorithm

The Discrete Case

Missing Data

The Continuous Case

EM and the KL Distance

Finite Mixture Problems

**Geometric Programming and the MART **Overview

An Example of a GP Problem

The Generalized AGM Inequality

Posynomials and the GP Problem

The Dual GP Problem

Solving the GP Problem

Solving the DGP Problem

Constrained Geometric Programming

Exercises

**Variational Inequality Problems and Algorithms **Monotone Functions

The Split-Feasibility Problem

The Variational Inequality Problem

Korpelevich’s Method for the VIP

On Some Algorithms of Noor

Split Variational Inequality Problems

Saddle Points

Exercises

**Set-Valued Functions in Optimization **Overview

Notation and Definitions

Basic Facts

Monotone Set-Valued Functions

Resolvents

Split Monotone Variational Inclusion

Solving the SMVIP

Special Cases of the SMVIP

The Split Common Null-Point Problem

Exercises

**Fenchel Duality**

The Legendre-Fenchel Transformation

Fenchel’s Duality Theorem

An Application to Game Theory

Exercises

**Compressed Sensing**Compressed Sensing

Sparse Solutions

Minimum One-Norm Solutions

Why Sparseness?

Compressed Sampling

**Appendix: Bregman-Legendre Functions **Essential Smoothness and Essential Strict Convexity

Bregman Projections onto Closed Convex Sets

Bregman-Legendre Functions

Bibliography

Index

### Biography

Charles Byrne