Bound-constrained polynomial optimization using only elementary calculations

Abstract

We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) 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 $f^H_k$ have a rate of convergence in $O(1/\sqrt {k})$. 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 $f^H_k$ needs only $O(n^k)$ elementary calculations.