Bound-Constrained Polynomial Optimization Using Only Elementary Calculations

We provide a monotone non increasing sequence of upper bounds f k (k ≥ 1) converging to the global minimum of a polynomial f on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim. 21, pp. 864–885, 2010] is that only elementary computations are required. For optimization over the hypercube, we show that the new bounds f k have a rate of convergence in O(1/ √ k). 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(k) function evaluations, while the new bound f k needs only O(n) elementary calculations.