Provides a self-contained, accessible introduction to the mathematical advances and challenges resulting from the use of semidefinite programming in polynomial optimization. This quickly evolving research area with contributions from the diverse fields of convex geometry, algebraic geometry, and optimization is known as convex algebraic geometry. Each chapter addresses a fundamental aspect of convex algebraic geometry. The book begins with an introduction to nonnegative polynomials and sums of squares and their connections to semidefinite programming and quickly advances to several areas at the forefront of current research. These include:* Semidefinite representability of convex sets.* Duality theory from the point of view of algebraic geometry.* Nontraditional topics such as sums of squares of complex forms and noncommutative sums of squares polynomials.
* List of Notation*
Chapter 1: What is Convex Algebraic Geometry?*
Chapter 2: Semidefinite Optimization*
Chapter 3: Polynomial Optimization, Sums of Squares, and Applications*
Chapter 4: Nonnegative Polynomials and Sums of Squares*
Chapter 5: Dualities*
Chapter 6: Semidefinite Representability*
Chapter 7: Convex Hulls of Algebraic Sets*
Chapter 8: Free Convexity*
Chapter 9: Sums of Hermitian Squares: Old and New* Appendix A: Background Material
Grigoriy Blekherman is an assistant professor at Georgia Institute of Technology and a 2012 recipient of the Sloan Research Fellowship. His research interests lie at the intersection of convex and algebraic geometry. Pablo A. Parrilo is a Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology. He has received the SIAG/CST Prize and the IEEE Antonio Ruberti Young Researcher Prize. His research interests include mathematical optimization, systems and control theory, and computational methods for engineering applications. Rekha R. Thomas is a Professor of Mathematics at the University of Washington. Her research interests are in optimization and computational algebra.
