This book provides a self-contained, accessible introduction to the mathematical advances and challenges resulting from the use of semidefinite programming in polynomial optimization. This important and highly applicable research area, with contributions from convex geometry, algebraic geometry and optimization, is known as convex algebraic geometry. Each chapter addresses a fundamental aspect of the topic, beginning with an introduction to nonnegative polynomials and sums of squares, and their connections to semidefinite programming. The material quickly advances to areas at the forefront of current research, including semidefinite representability of convex sets, duality theory in algebraic geometry, and nontraditional topics such as sums of squares of complex forms. The book is a suitable entry point to the subject for readers at the graduate level or above in mathematics, engineering or computer science. Instructors will find the book appropriate for a class or seminar, and researchers will encounter open problems and new research directions.
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.
List of contributors; List of figures; Preface; List of notation; 1. What is convex algebraic geometry? Grigoriy Blekherman, Pablo A. Parrilo and Rekha R. Thomas; 2. Semidefinite optimization Pablo A. Parrilo; 3. Polynomial optimization, sums of squares, and applications Pablo A. Parrilo; 4. Nonnegative polynomials and sums of squares Grigoriy Blekherman; 5. Dualities Philipp Rostalski and Bernd Sturmfels; 6. Semidefinite representability Jiawang Nie; 7. Convex hulls of algebraic sets Joao Gouveia and Rekha R. Thomas; 8. Free convexity J. William Helton, Igor Klep and Scott McCullough; 9. Sums of Hermitian squares: old and new Mihai Putinar; Appendix A. Background material Grigoriy Blekherman, Pablo A. Parrilo and Rekha R. Thomas; Index.
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
