We provide a monotone nonincreasing sequence of upper bounds fHk(k≥1)fkH(k≥1) converging to the global minimum of a polynomial f on simple sets like the unit hypercube in ℝn. 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 fHkfkH have a rate of convergence in O(1/k−−√)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(kn) function evaluations, while the new bound fHkfkH needs only O(nk) elementary calculations.
- polynomial optimization
- bound-constrained optimization
- Lasserre hierarchy
de Klerk, E., Lasserre, J. B., Laurent, M., & Sun, Z. (2017). Bound-constrained polynomial optimization using only elementary calculations. Mathematics of Operations Research, 42(3), 834-853. https://doi.org/10.1287/moor.2016.0829