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 Schmüdgen [Math. Ann., 289 (1991), pp. 203–206]. The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the hypercube.