Skip to main content
Log in

Quadrature-based polynomial optimization

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We show that Lasserre measure-based hierarchies for polynomial optimization can be implemented by directly computing the discrete minimum at a suitable set of algebraic quadrature nodes. The sampling cardinality can be much lower than in other approaches based on grids or norming meshes. All the vast literature on multivariate algebraic quadrature becomes in such a way relevant to polynomial optimization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bos, L., Caliari, M., De Marchi, S., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the generating curve approach. J. Approx. Theory 143, 15–25 (2006)

    Article  MathSciNet  Google Scholar 

  2. Caratheodory, C.: über den Variabilittsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)

    Article  Google Scholar 

  3. Cools, R.: Constructing cubature formulae: the science behind the art. Acta Numer. 6, 1–54 (1997)

    Article  MathSciNet  Google Scholar 

  4. Cools, R.: An encyclopaedia of cubature formulas. J. Complex. 19, 445–453 (2003)

    Article  MathSciNet  Google Scholar 

  5. de Klerk, E., Kuhn, D., Postek, K.: Distributionally robust optimization with polynomial densities: theory, models and algorithms. arXiv:1805.03588

  6. de Klerk, E., Lasserre, J.-B., Laurent, M., Sun, Z.: Bound-constrained polynomial optimization using only elementary calculations. Math. Oper. Res. 42, 834–853 (2017)

    Article  MathSciNet  Google Scholar 

  7. de Klerk, E., Laurent, M.: Comparison of Lasserre’s measure-based bounds for polynomial optimization to bounds obtained by simulated annealing. Math. Oper. Res. 43, 1317–1325 (2018)

    Article  MathSciNet  Google Scholar 

  8. de Klerk, E., Laurent, M.: Worst-case examples for Lasserre’s measure-based hierarchy for polynomial optimization on the hypercube. Math. Oper. Res. arXiv:1804.05524

  9. de Klerk, E., Hess, R., Laurent, M.: Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization. SIAM J. Optim. 27, 346–367 (2017)

    Article  MathSciNet  Google Scholar 

  10. de Klerk, E., Laurent, M., Sun, Z.: Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization. Math. Program. Ser. A 162(1), 363–392 (2017)

    Article  MathSciNet  Google Scholar 

  11. Jakeman, J.D., Narayan, A.: Generation and application of multivariate polynomial quadrature rules. Comput. Methods Appl. Mech. Eng. 338, 134–161 (2018)

    Article  MathSciNet  Google Scholar 

  12. Lasserre, J.B.: A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21, 864–885 (2011)

    Article  MathSciNet  Google Scholar 

  13. Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25(2), 185–200 (1975/76)

  14. Mousavi, S.E., Sukumar, N.: Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput. Mech. 47, 535–554 (2011)

    Article  MathSciNet  Google Scholar 

  15. Parlett, B.N.: The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, vol. 20. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  16. Piazzon, F., Sommariva, A., Vianello, M.: Caratheodory-Tchakaloff Least Squares, Sampling Theory and Applications. IEEE Xplore Digital Library (2017). https://doi.org/10.1109/SAMPTA.2017.8024337

  17. Piazzon, F., Vianello, M.: A note on total degree polynomial optimization by Chebyshev grids. Optim. Lett. 12, 63–71 (2018)

    Article  MathSciNet  Google Scholar 

  18. Piazzon, F., Vianello, M.: Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies. Optim. Lett. (2019). https://doi.org/10.1007/s11590-018-1377-0

  19. Putinar, M.: A note on Tchakaloff’s theorem. Proc. Am. Math. Soc. 125, 2409–2414 (1997)

    Article  MathSciNet  Google Scholar 

  20. Ryu, E.K., Boyd, S.P.: Extensions of Gauss quadrature via linear programming. Found. Comput. Math. 15, 953–971 (2015)

    Article  MathSciNet  Google Scholar 

  21. Sommariva, A., Vianello, M.: POLYGAUSS: a Matlab code for Gauss-like cubature over polygons. http://www.math.unipd.it/~alvise/software.html

  22. Sommariva, A., Vianello, M.: Compression of multivariate discrete measures and applications. Numer. Funct. Anal. Optim. 36, 1198–1223 (2015)

    Article  MathSciNet  Google Scholar 

  23. Sommariva, A., Vianello, M.: Nearly optimal nested sensors location for polynomial regression on complex geometries. Sampl. Theory Signal Image Process. 17, 95–101 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Tchakaloff, V.: Formules de cubature mécaniques à coefficients nonnégatifs. Bull. Sci. Math. 81, 123–134 (1957)

    MathSciNet  MATH  Google Scholar 

  25. Vianello, M.: Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization. Optim. Lett. 12, 1659–1667 (2018)

    Article  MathSciNet  Google Scholar 

  26. Xiao, H., Gimbutas, Z.: A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions. Comput. Math. Appl. 59, 663–676 (2010)

    Article  MathSciNet  Google Scholar 

  27. Xu, Y.: Minimal cubature rules and polynomial interpolation in two variables. J. Approx. Theory 164, 6–30 (2012)

    Article  MathSciNet  Google Scholar 

  28. Zhang, J.F., Kwong, C.P.: Some applications of a polynomial inequality to global optimization. J. Optim. Theory Appl. 127, 193–205 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Work partially supported by the DOR funds and the biennial project BIRD163015 of the University of Padova, and by the GNCS-INdAM. This research has been accomplished within the RITA “Research ITalian network on Approximation”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marco Vianello.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martinez, A., Piazzon, F., Sommariva, A. et al. Quadrature-based polynomial optimization. Optim Lett 14, 1027–1036 (2020). https://doi.org/10.1007/s11590-019-01416-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-019-01416-x

Keywords

Navigation