Semidefinite Optimization and Convex Algebraic Geometry

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 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.