Optimization of polynomials on compact semialgebraic sets

A basic closed semialgebraic subset S of R-n is defined by simultaneous polynomial inequalities g(1) >= 0,..., g(m) >= 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 in. mum 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 R-n converging to x*.