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Finite convergence and minimizer extraction in moment relaxations with correlative sparsity

Giovanni Fantuzzi, Federico Fuentes

TL;DR

This work delivers new, constructive conditions for finite convergence of moment-SOS relaxations applied to correlatively sparse polynomial optimization problems. By solving a correlatively sparse K-moment problem under running intersection and flat-extension-type rank conditions, the authors guarantee the existence of a finitely atomic representing measure and provide an explicit algorithm to extract multiple POP minimizers from the relaxation. The results generalize prior work by relaxing equal-rank and flat-overlap requirements, and they include measure-assembly methods that yield maximal-support atomic representations, as well as ways to obtain alternative representations via convex optimization. The approach is illustrated with practical examples showing finite convergence detection, minimizer recovery, and the necessity of the running intersection property, highlighting potential computational advantages over dense relaxations in structured sparsity settings.

Abstract

We identify a new sufficient condition for the finite convergence of moment relaxations of polynomial optimization problems with correlative sparsity. This condition, which follows from a solution to a correlatively sparse version of the classical truncated moment problem, requires that certain moment matrices admit a flat extension and that the variable cliques underpinning the relaxation satisfy a "running intersection" property. We also describe an algorithm that, when these conditions are met, extracts at least as many minimizers for the original polynomial optimization problem as the largest rank of the moment matrices in its relaxation. Our results, along with the necessity of the running intersection property, are illustrated with examples.

Finite convergence and minimizer extraction in moment relaxations with correlative sparsity

TL;DR

This work delivers new, constructive conditions for finite convergence of moment-SOS relaxations applied to correlatively sparse polynomial optimization problems. By solving a correlatively sparse K-moment problem under running intersection and flat-extension-type rank conditions, the authors guarantee the existence of a finitely atomic representing measure and provide an explicit algorithm to extract multiple POP minimizers from the relaxation. The results generalize prior work by relaxing equal-rank and flat-overlap requirements, and they include measure-assembly methods that yield maximal-support atomic representations, as well as ways to obtain alternative representations via convex optimization. The approach is illustrated with practical examples showing finite convergence detection, minimizer recovery, and the necessity of the running intersection property, highlighting potential computational advantages over dense relaxations in structured sparsity settings.

Abstract

We identify a new sufficient condition for the finite convergence of moment relaxations of polynomial optimization problems with correlative sparsity. This condition, which follows from a solution to a correlatively sparse version of the classical truncated moment problem, requires that certain moment matrices admit a flat extension and that the variable cliques underpinning the relaxation satisfy a "running intersection" property. We also describe an algorithm that, when these conditions are met, extracts at least as many minimizers for the original polynomial optimization problem as the largest rank of the moment matrices in its relaxation. Our results, along with the necessity of the running intersection property, are illustrated with examples.

Paper Structure

This paper contains 23 sections, 7 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Correlatively sparse POPs and their moment relaxations
  3. Main results
  4. Relation to previous work
  5. Preliminaries
  6. Correlatively sparse multi-indices and monomials
  7. Moment and localizing matrices
  8. Clique projections and pushforwards
  9. Proof of Theorem \ref{['thm:FF']}
  10. Clique subvectors are not zero
  11. Existence of the representing measure
  12. Recovering representing measures
  13. Construction of an atomic representing measure
  14. Maximality of the support
  15. Finding other atomic representing measures
  16. ...and 8 more sections

Key Result

Theorem 1.1

Fix a nonzero vector $\mathbf{y} =(y_\alpha)_{\alpha\in \mathcal{E}_{2\omega}}$. Assume that: Then, there exist an integer $r\geq \max_{i\in[m]} \mathop{\mathrm{rank}}\nolimits\mathbb{M}_{\Delta_{i}}^{\omega}(\mathbf{y})$, points $\mathbf{x}_1,\ldots,\mathbf{x}_r$ of $K$, and scalars $\lambda_1,\ldots,\lambda_r > 0$ such that the atomic measure $\lambda_1 \delta_{\mathbf{x}_1} + \cdots + \la

Figures (1)

  • Figure 5.1: Atom concatenation strategy for the example of \ref{['ss:ex-pop']}. An atom $(a,b)$ for $\mu_{\Delta_i}$ is concatenated with an atom $(c,d)$ for $\mu_{\Delta_{i+1}}$ if $b=c$. Each colored paths leads to a different minimizer for \ref{['e:ex-pop']}, which is an atom of the representing measure $\mu$ in \ref{['e:ex-atomic-mu']}.

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3: Lasserre2006
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 4 more