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Revisiting the convergence rate of the Lasserre hierarchy for polynomial optimization over the hypercube

Sander Gribling, Etienne de Klerk, Juan Vera

TL;DR

This work addresses minimizing a polynomial on the hypercube $[-1,1]^n$ via the Lasserre hierarchy and investigates the convergence rate of the resulting lower bounds. By developing a polynomial kernel method that yields Putinar-type certificates, the authors prove an $O(\log r / r^2)$ convergence rate to an $\varepsilon$-approximation for fixed $\varepsilon>0$, advancing beyond the known $O(1/r)$ rate and providing explicit degree and norm bounds. The paper also establishes fundamental limits of this kernel-approximation approach through Stengle-type lower bounds and supports the analysis with numerical experiments on kernel coefficients. Overall, the results clarify both the potential of kernel-based SOS certificates to tighten convergence and the inherent barriers faced by this technique, suggesting directions for achieving tighter rates or alternative hierarchies.

Abstract

We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the minimum value $f^*$, where $r$ refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter $r$. The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order $1/r$. Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order $1/r^2$ for our approach.

Revisiting the convergence rate of the Lasserre hierarchy for polynomial optimization over the hypercube

TL;DR

This work addresses minimizing a polynomial on the hypercube via the Lasserre hierarchy and investigates the convergence rate of the resulting lower bounds. By developing a polynomial kernel method that yields Putinar-type certificates, the authors prove an convergence rate to an -approximation for fixed , advancing beyond the known rate and providing explicit degree and norm bounds. The paper also establishes fundamental limits of this kernel-approximation approach through Stengle-type lower bounds and supports the analysis with numerical experiments on kernel coefficients. Overall, the results clarify both the potential of kernel-based SOS certificates to tighten convergence and the inherent barriers faced by this technique, suggesting directions for achieving tighter rates or alternative hierarchies.

Abstract

We revisit the problem of minimizing a given polynomial on the hypercube . Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds on the minimum value , where refers to the allowed degrees in the sum-of-squares hierarchy. A natural question is how fast the hierarchy converges as a function of the parameter . The current state-of-the-art is due to Baldi and Slot [SIAM J. on Applied Algebraic Geometry, 2024] and roughly shows a convergence rate of order . Here we obtain closely related results via a different approach: the polynomial kernel method. We also discuss limitations of the polynomial kernel method, suggesting a lower bound of order for our approach.
Paper Structure (15 sections, 34 theorems, 124 equations, 3 figures, 1 table)

This paper contains 15 sections, 34 theorems, 124 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of Chebyshev polynomials
  4. Technical lemmas related to quadratic modules
  5. An approximation kernel approach
  6. Kernels of the Gaussian type
  7. Truncated Gaussian kernel
  8. An SOS kernel
  9. Proof of main result
  10. Proof in the univariate case
  11. Extension to the multivariate setting
  12. Limits to our proof technique
  13. Bounds based on examples by Stengle
  14. Numerical investigation
  15. Conclusion and discussion

Key Result

Theorem 1

Assume a compact semi-algebraic set $K$ is described by polynomials $\mathbf{g}$ and that the Archimedean assumption Archimedean_property holds for $\mathcal{Q}(\mathbf{g})$. Then, if $f \in \mathbb{R}[x]$ is (strictly) positive on $K$, one has $f \in \mathcal{Q}(\mathbf{g})$.

Figures (3)

  • Figure 1: Plots of the values $1/v_{r,d}$ as a function of $r$ for fixed $d \in \{1,2,3,4\}$.
  • Figure 2: Plots of the values $v_{r,d}\cdot (r/d)$ as a function of $r$ for fixed $d \in \{1,2,3,4\}$.
  • Figure 3: Plots of the values $v_{r,d}\cdot (r/d)^2$ as a function of $r$ for fixed $d \in \{1,2,3,4\}$.

Theorems & Definitions (58)

  • Theorem 1: Putinar Positivstellensatz Putinar1993
  • Theorem 2: Theorem 11 in Baldi and Slot doi:10.1137/23M1555430
  • Corollary 1: cf. Corollary 15 in Baldi and Slot doi:10.1137/23M1555430
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 48 more