Bound-constrained polynomial optimization using only elementary calculations

TL;DR

This paper introduces a sequence of upper bounds for polynomial optimization on simple sets like the hypercube, requiring only elementary calculations and achieving competitive convergence rates.

Contribution

It presents a new method for polynomial optimization that uses elementary computations and converges at known rates, improving practicality over previous approaches.

Findings

Converges at rate O(1/√k) for general polynomials.

Achieves rate O(1/k) for quadratic polynomials and those with rational minimizers.

Requires only O(n^k) elementary calculations, less than grid evaluation methods.

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.