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Rank conditions for exactness of semidefinite relaxations in polynomial optimization

Jean B Lasserre

TL;DR

This work analyzes finite convergence in the Moment-SOS hierarchy for polynomial optimization by deriving a practical rank-based condition on the moment matrix. If $n\ge v$ and $s=\operatorname{rank}(\mathbf{M}_n(\boldsymbol{\phi}^n))\le n-v+1$, the truncated $\mathbf{K}$-moment problem admits a representing measure supported on at most $r\le s$ points in $\mathbf{K}$, and if $r=s$, the full moment sequence has a representing measure. Consequently, an optimal SDP relaxation yields $\rho_n=f^*$ (exactness) and, in the QCQP setting, enables extraction of global minimizers; the results localize the representing measure to the feasible set, addressing limitations of unconstrained rank criteria such as Blekherman’s. The paper also discusses the unconstrained case via homogenization, showing that a rank bound on the homogenized moment matrix can certify exactness of a single SDP relaxation for unconstrained POPs. Together, these rank-based criteria provide practical stopping rules and minimizer extraction strategies for both constrained and unconstrained polynomial optimization problems.

Abstract

We consider the Moment-SOS hierarchy in polynomial optimization. We first provide a sufficient condition to solve the truncated K-moment problem associated with a given degree-$2n$ pseudo-moment sequence $φ$ n and a semi-algebraic set $K \subset \mathbb{R}^d$. Namely, let $2v$ be the maximum degree of the polynomials that describe $K$. If the rank $r$ of its associated moment matrix is less than $nv + 1$, then $φ^n$ has an atomic representing measure supported on at most $r$ points of $K$. When used at step-$n$ of the Moment-SOS hierarchy, it provides a sufficient condition to guarantee its finite convergence (i.e., the optimal value of the corresponding degree-n semidefinite relaxation of the hierarchy is the global minimum). For Quadratic Constrained Quadratic Problems (QCQPs) one may also recover global minimizers from the optimal pseudo-moment sequence. Our condition is in the spirit of Blekherman's rank condition and while on the one-hand it is more restrictive, on the other hand it applies to constrained POPs as it provides a localization on $K$ for the representing measure.

Rank conditions for exactness of semidefinite relaxations in polynomial optimization

TL;DR

This work analyzes finite convergence in the Moment-SOS hierarchy for polynomial optimization by deriving a practical rank-based condition on the moment matrix. If and , the truncated -moment problem admits a representing measure supported on at most points in , and if , the full moment sequence has a representing measure. Consequently, an optimal SDP relaxation yields (exactness) and, in the QCQP setting, enables extraction of global minimizers; the results localize the representing measure to the feasible set, addressing limitations of unconstrained rank criteria such as Blekherman’s. The paper also discusses the unconstrained case via homogenization, showing that a rank bound on the homogenized moment matrix can certify exactness of a single SDP relaxation for unconstrained POPs. Together, these rank-based criteria provide practical stopping rules and minimizer extraction strategies for both constrained and unconstrained polynomial optimization problems.

Abstract

We consider the Moment-SOS hierarchy in polynomial optimization. We first provide a sufficient condition to solve the truncated K-moment problem associated with a given degree- pseudo-moment sequence n and a semi-algebraic set . Namely, let be the maximum degree of the polynomials that describe . If the rank of its associated moment matrix is less than , then has an atomic representing measure supported on at most points of . When used at step- of the Moment-SOS hierarchy, it provides a sufficient condition to guarantee its finite convergence (i.e., the optimal value of the corresponding degree-n semidefinite relaxation of the hierarchy is the global minimum). For Quadratic Constrained Quadratic Problems (QCQPs) one may also recover global minimizers from the optimal pseudo-moment sequence. Our condition is in the spirit of Blekherman's rank condition and while on the one-hand it is more restrictive, on the other hand it applies to constrained POPs as it provides a localization on for the representing measure.
Paper Structure (10 sections, 8 theorems, 40 equations)

This paper contains 10 sections, 8 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. On rank conditions
  3. Contribution
  4. Notation, definitions and preliminary results
  5. Main result
  6. The unconstrained case
  7. Conclusion
  8. Appendix
  9. Proof of Theorem \ref{['th1']}
  10. Proof of Theorem \ref{['th-main']}

Key Result

Theorem 1.2

With $n\geq v$, let $\boldsymbol{\phi}^n=(\phi_{\boldsymbol{\alpha}})_{\boldsymbol{\alpha}\in\mathbb{N}^d_{2n}}$ be such that If $s:=\mathrm{rank}(\mathbf{M}_n(\boldsymbol{\phi}^n))\,\leq\,n-v+1$ then $\boldsymbol{\phi}^{\hat{n}}:=(\phi^n_{\boldsymbol{\alpha}})_{\boldsymbol{\alpha}\in\mathbb{N}^d_{2n-1}}$ has a representing measure supported on at most $r\,(\leq s)$ points of $\mathbf{K}$. In add

Theorems & Definitions (15)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • ...and 5 more