We provide a monotone nonincreasing sequence of upper bounds [Formula: see text] converging to the global minimum of a polynomial f on simple sets like the unit hypercube in ℝ
. The novelty with respect to the converging sequence of upper bounds in Lasserre [Lasserre JB (2010) A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim. 21:864–885] is that only elementary computations are required. For optimization over the hypercube [0, 1]
, we show that the new bounds [Formula: see text] have a rate of convergence in [Formula: see text]. Moreover, we show a stronger convergence rate in O(1/k) for quadratic polynomials and more generally for polynomials having a rational minimizer in the hypercube. In comparison, evaluation of all rational grid points with denominator k produces bounds with a rate of convergence in O(1/k2
), but at the cost of O(kn
) function evaluations, while the new bound [Formula: see text] needs only O(nk
) elementary calculations.