  In this survey we consider polynomial optimization problems, asking to
minimize a polynomial function over a compact semialgebraic set, defined by
polynomial inequalities. This models a great variety of (in general, nonlinear
nonconvex) optimization problems. Various hierarchies of (lower and upper)
bounds have been introduced, having the remarkable property that they converge
asymptotically to the global minimum. These bounds exploit algebraic
representations of positive polynomials in terms of sums of squares and can be
computed using semidefinite optimization. Our focus lies in the performance
analysis of these hierarchies of bounds, namely, in how far the bounds are from
the global minimum as the degrees of the sums of squares they involve tend to
infinity. We present the main state-of-the-art results and offer a gentle
introductory overview over the various techniques that have been recently
developed to establish them, stemming from the theory of orthogonal
polynomials, approximation theory, Fourier analysis, and more.
