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.
Similar content being viewed by others
References
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)
Caratheodory, C.: über den Variabilittsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)
Cools, R.: Constructing cubature formulae: the science behind the art. Acta Numer. 6, 1–54 (1997)
Cools, R.: An encyclopaedia of cubature formulas. J. Complex. 19, 445–453 (2003)
de Klerk, E., Kuhn, D., Postek, K.: Distributionally robust optimization with polynomial densities: theory, models and algorithms. arXiv:1805.03588
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)
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)
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
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)
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)
Jakeman, J.D., Narayan, A.: Generation and application of multivariate polynomial quadrature rules. Comput. Methods Appl. Mech. Eng. 338, 134–161 (2018)
Lasserre, J.B.: A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21, 864–885 (2011)
Möller, H.M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math. 25(2), 185–200 (1975/76)
Mousavi, S.E., Sukumar, N.: Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput. Mech. 47, 535–554 (2011)
Parlett, B.N.: The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, vol. 20. SIAM, Philadelphia (1998)
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
Piazzon, F., Vianello, M.: A note on total degree polynomial optimization by Chebyshev grids. Optim. Lett. 12, 63–71 (2018)
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
Putinar, M.: A note on Tchakaloff’s theorem. Proc. Am. Math. Soc. 125, 2409–2414 (1997)
Ryu, E.K., Boyd, S.P.: Extensions of Gauss quadrature via linear programming. Found. Comput. Math. 15, 953–971 (2015)
Sommariva, A., Vianello, M.: POLYGAUSS: a Matlab code for Gauss-like cubature over polygons. http://www.math.unipd.it/~alvise/software.html
Sommariva, A., Vianello, M.: Compression of multivariate discrete measures and applications. Numer. Funct. Anal. Optim. 36, 1198–1223 (2015)
Sommariva, A., Vianello, M.: Nearly optimal nested sensors location for polynomial regression on complex geometries. Sampl. Theory Signal Image Process. 17, 95–101 (2018)
Tchakaloff, V.: Formules de cubature mécaniques à coefficients nonnégatifs. Bull. Sci. Math. 81, 123–134 (1957)
Vianello, M.: Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization. Optim. Lett. 12, 1659–1667 (2018)
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)
Xu, Y.: Minimal cubature rules and polynomial interpolation in two variables. J. Approx. Theory 164, 6–30 (2012)
Zhang, J.F., Kwong, C.P.: Some applications of a polynomial inequality to global optimization. J. Optim. Theory Appl. 127, 193–205 (2005)
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
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-019-01416-x