Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel--Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.

MSC codes

  1. polynomial optimization
  2. Positivstellensatz
  3. sum-of-squares hierarchy
  4. Christoffel--Darboux kernel
  5. polynomial kernel method

MSC codes

  1. 90C22
  2. 90C23
  3. 90C26

Get full access to this article

View all available purchase options and get full access to this article.


L. Baldi and B. Mourrain, On the Effective Putinar's Positivstellensatz and Moment Approximation, preprint, https://arxiv.org/abs/2111.11258, 2021.
Y. de Castro, F. Gamboa, D. Henrion, and J. B. Lasserre, Dual optimal design and the Christoffel-Darboux polynomial, Optim. Lett., 15 (2021), pp. 3--8.
E. de Klerk, R. Hess, and M. Laurent, Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization, SIAM J. Optim., 27 (2017), pp. 347--367, https://doi.org/10.1137/16M1065264.
E. de Klerk and M. Laurent, Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube, SIAM J. Optim., 20 (2010), pp. 3104--3120, https://doi.org/10.1137/100790835.
E. de Klerk and M. Laurent, Comparison of Lasserre's measure-based bounds for polynomial optimization to bounds obtained by simulated annealing, Math. Oper. Res., 43 (2018), pp. 1317--1325.
E. de Klerk and M. Laurent, A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis, in World Women in Mathematics 2018: Proceedings of the First World Meeting for Women in Mathematics (WM), C. Araujo, G. Benkart, C. E. Praeger, and B. Tanbay, eds., Springer, Cham, 2019, pp. 17--56.
E. de Klerk and M. Laurent, Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube, Math. Oper. Res., 45 (2020), pp. 86--98.
E. de Klerk and M. Laurent, Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere, Math. Program., 193 (2022), pp. 665--685.
E. de Klerk, M. Laurent, and Z. Sun, Convergence analysis for Lasserre's measure-based hierarchy of upper bounds for polynomial optimization, Math. Program., 162 (2017), pp. 363--392.
K. Fang and H. Fawzi, The sum-of-squares hierarchy on the sphere and applications in quantum information theory, Math. Program., 190 (2021), pp. 331--360.
H. Fawzi, J. Saunderson, and P. A. Parrilo, Sparse sums of squares on finite abelian groups and improved semidefinite lifts, Math. Program., 160 (2016), pp. 149--191.
F. Kirschner and E. de Klerk, Convergence Rates of RLT and Lasserre-type Hierarchies for the Generalized Moment Problem over the Simplex and the Sphere, preprint, https://arxiv.org/abs/2103.02924, 2021.
J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796--817, https://doi.org/10.1137/S1052623400366802.
J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010.
J. B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization, SIAM J. Optim., 21 (2011), pp. 864--885, https://doi.org/10.1137/100806990.
J. B. Lasserre, The moment-SOS hierarchy and the Christoffel-Darboux kernel, Optim. Lett., 15 (2021), pp. 1835--1845.
J. B. Lasserre and E. Pauwels, The empirical Christoffel function with applications in data analysis, Adv. Comput. Math., 45 (2019), pp. 1439--1468.
M. Laurent, Sums of squares, moment matrices and optimization over polynomials, in Emerging Applications of Algebraic Geometry, IMA Vol. Math. Appl. 149, Springer, New York, 2009, pp. 157--270, https://doi.org/10.1007/978-0-387-09686-5_7.
M. Laurent and L. Slot, An Effective Version of Schmüdgen's Positivstellensatz for the Hypercube, preprint, https://arxiv.org/abs/2109.09528, 2021; Optim. Lett. submitted.
N. H. A. Mai and V. Magron, On the Complexity of Putinar-Vasilescu's Positivstellensatz, preprint, https://arxiv.org/abs/2104.11606, 2021.
S. Marx, E. Pauwels, T. Weisser, D. Henrion, and J. B. Lasserre, Semi-algebraic approximation using Christoffel--Darboux kernel, Constr. Approx., 54 (2021), pp. 391--429.
J. Nie and M. Schweighofer, On the complexity of Putinar's Positivstellensatz, J. Complexity, 23 (2007), pp. 135--150.
E. Pauwels, M. Putinar, and J. B. Lasserre, Data analysis from empirical moments and the Christoffel function, Found. Comput. Math., 21 (2021), pp. 243--273.
S. Sakaue, A. Takeda, S. Kim, and N. Ito, Exact semidefinite programming relaxations with truncated moment matrix for binary polynomial optimization problems, SIAM J. Optim., 27 (2017), pp. 565--582, https://doi.org/10.1137/16M105544X.
M. Schweighofer, On the complexity of Schmüdgen's Positivstellensatz, J. Complexity, 20 (2004), pp. 529--543.
L. Slot and M. Laurent, Near optimal analysis of Lasserre's univariate measure-based bounds for multivariate polynomial optimization, Math. Program., 188 (2021), pp. 443--460.
L. Slot and M. Laurent, Sum-of-squares hierarchies for binary polynomial optimization, in Integer Programming and Combinatorial Optimization, M. Singh and D. P. Williamson, eds., Lecture Notes in Comput. Sci. 12707, Springer, Cham, 2021, pp. 43--57, https://doi.org/10.1007/s10107-021-01745-9.
L. Slot and M. Laurent, Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets, Math. Program., 193 (2022), pp. 831--871, https://doi.org/10.1007/s10107-020-01468-3.
G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, RI, 1975.
V. Tchakaloff, Formules de cubature mecanique coefficients non négatifs, Bull. Sci. Math., 81 (1957), pp. 123--134.
A. Weisse, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys., 78 (2006), pp. 275--306.
Y. Xu, Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in $\mathbb{R}^d$, Proc. Amer. Math. Soc., 126 (1998), pp. 3027--3036, http://www.jstor.org/stable/119105.
Y. Xu, Summability of Fourier orthogonal series for Jacobi weight on a ball in $\mathbb{R}^d$, Trans. Amer. Math. Soc., 351 (1999), pp. 2439--2458, http://www.jstor.org/stable/117878.

Information & Authors


Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 2612 - 2635
ISSN (online): 1095-7189


Submitted: 9 November 2021
Accepted: 3 May 2022
Published online: 3 November 2022

MSC codes

  1. polynomial optimization
  2. Positivstellensatz
  3. sum-of-squares hierarchy
  4. Christoffel--Darboux kernel
  5. polynomial kernel method

MSC codes

  1. 90C22
  2. 90C23
  3. 90C26



Funding Information

H2020 Marie Skłodowska-Curie Actions https://doi.org/10.13039/100010665 : 764759

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media