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Density, Determinacy, Duality and a Regularized Moment-SOS Hierarchy

Didier Henrion

TL;DR

The paper extends the moment-SOS framework by introducing a regularized hierarchy that does not rely on Putinar’s Positivstellensatz or Archimedean properties, enabling global polynomial optimization on unbounded or non-Archimedean sets. It leverages the multivariate Carleman condition to establish density of polynomials, SOS, and quadratic modules in L^2(μ), and constructs a convergent dual SOS and primal pseudo-moment formulation within a function-space viewpoint. A Bernstein–Markov inequality-based penalized variant yields monotone, certified lower bounds that converge to the global optimum, with the standard hierarchy recovered in the appropriate limit. The approach is demonstrated on challenging benchmark problems (Motzkin, Origin, Stengle, Prestel–Delzell), illustrating convergence rates and the practical impact of BM constants. Together, these results broaden the applicability of SOS-based global optimization to unbounded and non-Archimedean settings while preserving convergence guarantees.

Abstract

The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar's Positivstellensatz, which requires the feasible set to satisfy the algebraic Archimedean property. In this paper, we introduce a regularized moment-SOS hierarchy capable of handling problems on unbounded sets or bounded sets violating the Archimedean property. Adopting a functional analysis viewpoint, we rely on the multivariate Carleman condition for measure determinacy rather than algebraic compactness. We prove that finite degree projections of the quadratic module are dense in the cone of positive polynomials with respect to the square norm induced by the measure. Based on these density results, we prove the convergence of a regularized hierarchy without invoking any Positivstellensatz. Furthermore, we propose a penalized formulation of the hierarchy which, combined with Bernstein-Markov inequalities, provides a monotonically non-decreasing sequence of certified lower bounds on the global minimum. The approach is illustrated on several benchmark problems known to be difficult or ill-posed for the standard hierarchy.

Density, Determinacy, Duality and a Regularized Moment-SOS Hierarchy

TL;DR

The paper extends the moment-SOS framework by introducing a regularized hierarchy that does not rely on Putinar’s Positivstellensatz or Archimedean properties, enabling global polynomial optimization on unbounded or non-Archimedean sets. It leverages the multivariate Carleman condition to establish density of polynomials, SOS, and quadratic modules in L^2(μ), and constructs a convergent dual SOS and primal pseudo-moment formulation within a function-space viewpoint. A Bernstein–Markov inequality-based penalized variant yields monotone, certified lower bounds that converge to the global optimum, with the standard hierarchy recovered in the appropriate limit. The approach is demonstrated on challenging benchmark problems (Motzkin, Origin, Stengle, Prestel–Delzell), illustrating convergence rates and the practical impact of BM constants. Together, these results broaden the applicability of SOS-based global optimization to unbounded and non-Archimedean settings while preserving convergence guarantees.

Abstract

The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar's Positivstellensatz, which requires the feasible set to satisfy the algebraic Archimedean property. In this paper, we introduce a regularized moment-SOS hierarchy capable of handling problems on unbounded sets or bounded sets violating the Archimedean property. Adopting a functional analysis viewpoint, we rely on the multivariate Carleman condition for measure determinacy rather than algebraic compactness. We prove that finite degree projections of the quadratic module are dense in the cone of positive polynomials with respect to the square norm induced by the measure. Based on these density results, we prove the convergence of a regularized hierarchy without invoking any Positivstellensatz. Furthermore, we propose a penalized formulation of the hierarchy which, combined with Bernstein-Markov inequalities, provides a monotonically non-decreasing sequence of certified lower bounds on the global minimum. The approach is illustrated on several benchmark problems known to be difficult or ill-posed for the standard hierarchy.

Paper Structure

This paper contains 15 sections, 17 theorems, 116 equations, 1 figure.

Table of Contents

  1. Carleman Determinacy
  2. Polynomial Density
  3. SOS Density
  4. Quadratic Module Density
  5. A regularized moment-SOS hierarchy
  6. Dual SOS problem
  7. Primal pseudo-moment problem
  8. Convergence
  9. Lower bounds with penalized formulation
  10. Recovering the standard moment-SOS hierarchy
  11. Examples
  12. Motzkin
  13. Origin
  14. Stengle
  15. Prestel and Delzell

Key Result

Theorem 1

If $\mu$ satisfies Carleman, then it is determinate.

Figures (1)

  • Figure 1: Log-log plot of the gaps $|p_d^*-p^*|$ versus the relaxation order $d$ for the Stengle POP. The near-linearity indicates an algebraic decay $|p_d^*-p^*|\approx C\,d^{-2}$.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 34 more