Bound-constrained polynomial optimization using only elementary calculations

Etienne De Klerk, Jean B. Lasserre, Monique Laurent, Zhao Sun

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Abstract

We provide a monotone nonincreasing sequence of upper bounds f H k (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 f H k  fkH 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 f H k  fkH needs only O(nk) elementary calculations.

Original languageEnglish
Pages (from-to)834-853
Number of pages20
JournalMathematics of Operations Research
Volume42
Issue number3
DOIs
Publication statusPublished - 15 Mar 2017

Keywords

  • Bound-constrained optimization
  • Lasserre hierarchy
  • Polynomial optimization

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