Bound-Constrained Polynomial Optimization Using Only Elementary Calculations

We provide a monotone nonincreasing sequence of upper bounds f k H ( k ≥ 1 ) converging to the global minimum of a polynomial f on simple sets like the unit hypercube in ℝ n . 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] n , we show that the new bounds f k H 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/ k 2 ), but at the cost of O ( k n ) function evaluations, while the new bound f k H needs only O ( n k ) elementary calculations.