Sums of Squares and Optimization

Sums of squares techniques have become an indispensable tool in polynomial and semidefinite optimization. Chapter 8 offers an introduction to some of the most important concepts and results. After an overview of important general notions from convexity, spectrahedra and their shadows are introduced. The moment relaxation approach to polynomial optimization is discussed in detail, it rests on various positivstellensätze from previous chapters. The last part addresses the characterization of spectrahedral shadows, which are the feasible sets of semidefinite programming. Results by Helton–Nie guarantee the existence of semidefinite representations under fairly general conditions. Constructions due to the author show, on the other hand, that prominent convex sets do not allow such a representation.