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We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge1$) converging to the global minimum of a polynomial $f$ on simple sets likethe unit hypercube. The novelty with respect to the converging sequence ofupper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets andpolynomial optimization, SIAM J. Optim. 21, pp. 864--885, 2010] is that onlyelementary computations are required. For optimization over the hypercube, weshow that the new bounds $f^H_k$ have a rate of convergence in $O(1/\sqrt{k})$. Moreover we show a stronger convergence rate in $O(1/k)$ for quadraticpolynomials and more generally for polynomials having a rational minimizer inthe hypercube. In comparison, evaluation of all rational grid points withdenominator $k$ produces bounds with a rate of convergence in $O(1/k^2)$, butat the cost of $O(k^n)$ function evaluations, while the new bound $f^H_k$ needsonly $O(n^k)$ elementary calculations.

Bound-constrained polynomial optimization using only elementary calculations