Sum of Squares Programs and Polynomial Inequalities

How can one find real solutions (x1, x2)? How to prove that they do not exist? And if the solution set is nonempty, how to optimize a polynomial function over this set? Until a few years ago, the default answer to these and similar questions would have been that the possi­ ble nonconvexity of the feasible set and/or objective function precludes any kind of analytic global results. Even today, the methods of choice for most prac­ titioners would probably employ mostly local tech­ niques (Newton’s and its variations), possibly com­ plemented by a systematic search using deterministic or stochastic exploration of the solution space, inter­ val analysis or branch and bound. However, very recently there have been renewed hopes for the efficient solution of specific instances of this kind of problems. The main reason is the appear­ ance of methods that combine in a very interesting fashion ideas from real algebraic geometry and convex optimization [27, 30, 21]. As we will see, these meth­ ods are based on the intimate links between sum of squares decompositions for multivariate polynomials and semidefinite programming (SDP). In this note we outline the essential elements of this new research approach as introduced in [30, 32], and provide pointers to the literature. The center­ pieces will be the following two facts about multi­ variate polynomials and systems of polynomials in­ equalities: Sum of squares decompositions can be com­ puted using semidefinite programming.