Sum of Squares: Theory and Applications

This book provides a comprehensive overview of the theory and applications of polynomials that are sums of squares,with rich connections to various areas of mathematics and practical applications.

Format: Paperback / softback
Length: 142 pages
Publication date: 30 September 2020
Publisher: American Mathematical Society

Introduction:
This volume is a comprehensive and up-to-date exploration of the theory and applications of polynomials that are sums of squares. It offers a concise yet comprehensive overview of this exciting and timely topic,with rich connections to various branches of mathematics,including polynomial and semidefinite optimization,real and convex algebraic geometry,and theoretical computer science.
Chapter Summaries:
The six chapters of this book introduce and survey recent developments in the field of sums of squares. These chapters cover a wide range of topics,including the algebraic and geometric aspects of sums of squares and spectrahedra,lifted representations of convex sets,and the algorithmic and computational implications of viewing sums of squares as a meta algorithm.
In Chapter 1, the authors provide an introduction to sums of squares and their historical background. They discuss the fundamental properties of sums of squares and their connections to other areas of mathematics,such as polynomial optimization and semidefinite programming. The chapter also highlights the practical applications of sums of squares in various fields,such as control theory,statistics,finance,and machine learning.
Chapter 2 explores the algebraic and geometric aspects of sums of squares. The authors discuss the properties of sums of squares and their connections to other areas of mathematics,such as convex optimization,semidefinite programming,and algebraic geometry. They also introduce the concept of lifted representations of convex sets and their applications in the study of sums of squares.
Chapter 3 focuses on the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The authors discuss the advantages and disadvantages of this approach and introduce various algorithms and techniques for solving sums of squares,including convex optimization algorithms,semidefinite programming algorithms,and matrix decomposition methods.
Chapter 4 discusses the practical applications of sums of squares across a variety of areas. The authors showcase examples from control theory,statistics,finance,and machine learning,where sums of squares are used to solve optimization problems,estimate probabilities,and develop machine learning models.
Chapter 5 explores the connections between sums of squares and other areas of mathematics,such as semidefinite programming,convex optimization,and algebraic geometry. The authors discuss the recent developments in these areas and their implications for the study of sums of squares.
Chapter 6 concludes the book by summarizing the key findings and future directions in the study of sums of squares. The authors suggest potential research topics and open questions that need to be addressed in the future.
Conclusion:
In conclusion, this volume provides a comprehensive and state-of-the-art overview of the theory and applications of polynomials that are sums of squares. It offers a valuable resource for researchers,students,and practitioners in mathematics,computer science,and related fields. The book showcases practical applications of the techniques across a variety of areas,making it an essential tool for anyone interested in developing new algorithms and models for solving optimization problems.

Weight: 278g
Dimension: 182 x 257 x 12 (mm)
ISBN-13: 9781470450250