Sum of squares : theory and applications : AMS short course, sum of squares : theory and applications, January 14-15, 2019, Baltimore, Maryland

This volume is based on lectures delivered at the 2019 AMS Short Course Sum of Squares: Theory and Applications , held January 14 15, 2019, in Baltimore, Maryland. This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science. The six chapters introduce and survey recent developments in this area; specific topics include 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. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning. Cover Title page Contents Preface A brief introduction to sums of squares 1. Two guiding questions 2. Finding sum of squares decompositions 3. Nonnegative polynomials and sums of squares 4. Sums of squares and optimization 5. Adding constraints References The geometry of spectrahedra 1. Introduction 2. Background 3. Convex duality 4. Facial structure, extreme points and ranks 5. Moment problems and sums of squares References Lifts of convex sets 1. Introduction 2. Linear programming lifts and Yannakakis’ theorem 3. Semidefinite programming lifts, positive semidefinite factorizations, and sums of squares 4. Conclusion 5. Summary References Algebraic geometry and sums of squares 1. Introduction 2. Projective geometry and sums of squares. 3. Main Theorem and the classification of varieties of minimal degree. References Sums of squares in theoretical computer science 1. MAXCUT 2. Rounding SOS for MAXCUT 3. Rounding Higher Degree SOS 4. Degree Lower Bounds for Constraint Satisfaction Problems 5. Constructing a Pseudo-Expectation 6. Discussion References Applications of sums of squares Part 1. Dynamical systems and control 1. Certifying properties of a polynomial dynamical system 2. Stability of switched linear systems 3. Related applications Part 2. Probability and measure theory 4. The moment problem 5. Probability bounds given moments of a random variable and applications to option pricing Part 3. Statistics and machine learning 6. Shape-constrained regression 7. Optimal design Part 4. Other applications 8. Optimization 9. Game theory: polynomial games Part 5. Conclusion: a word on implementation challenges Acknowledgments References Back Cover