Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube

We consider the problem of minimizing a polynomial on the hypercube [0, 1]n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmu ̈dgen [26]. The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the hypercube.