Optimization of Polynomials on Compact Semialgebraic Sets

A basic closed semialgebraic subset S of ⁿ is defined by simultaneous polynomial inequalities g₁≥ 0,,gₘ≥ 0. We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f^ of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x^, we prove that every sequence of "nearly" optimal solutions of the successive relaxations gives rise to a sequence of points in ⁿ converging to x^.