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A Characterization for Tightness of the Sparse Moment-SOS Hierarchy

Jiawang Nie, Zheng Qu, Xindong Tang, Linghao Zhang

TL;DR

The paper addresses tightness of the sparse Moment-SOS hierarchy for sparse polynomial optimization and provides a precise necessary-and-sufficient condition for tightness in terms of a sum-of-polynomials decomposition relative to sparse ideals and quadratic modules. It extends the dense Moment-SOS theory by incorporating correlative sparsity, the running intersection property, and sparse Positivstellensatz certificates, along with a sparse flat-truncation method to detect tightness and extract minimizers. The authors present several broad-scope sufficient conditions for tightness, including convexity, sufficient optimality conditions, and finiteness of constraining sets, as well as Schmüdgen-type sparse relaxations, with supporting examples and proofs. The results offer practical criteria for certifying finite convergence and devising minimizer extraction algorithms in large-scale sparse polynomial optimization, providing both theoretical and computational insights into when sparse relaxations suffice.

Abstract

This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.

A Characterization for Tightness of the Sparse Moment-SOS Hierarchy

TL;DR

The paper addresses tightness of the sparse Moment-SOS hierarchy for sparse polynomial optimization and provides a precise necessary-and-sufficient condition for tightness in terms of a sum-of-polynomials decomposition relative to sparse ideals and quadratic modules. It extends the dense Moment-SOS theory by incorporating correlative sparsity, the running intersection property, and sparse Positivstellensatz certificates, along with a sparse flat-truncation method to detect tightness and extract minimizers. The authors present several broad-scope sufficient conditions for tightness, including convexity, sufficient optimality conditions, and finiteness of constraining sets, as well as Schmüdgen-type sparse relaxations, with supporting examples and proofs. The results offer practical criteria for certifying finite convergence and devising minimizer extraction algorithms in large-scale sparse polynomial optimization, providing both theoretical and computational insights into when sparse relaxations suffice.

Abstract

This paper studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.
Paper Structure (21 sections, 16 theorems, 167 equations, 2 tables)

This paper contains 21 sections, 16 theorems, 167 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideals and SOS polynomials
  4. Sparse moments
  5. The characterization for tightness
  6. Characterization for the tightness
  7. Detecting tightness and extracting minimizers
  8. Some sufficient conditions for tightness
  9. The convex case
  10. The case of sufficient optimality conditions
  11. The case of finite sets
  12. The Schmüdgen type sparse relaxations
  13. Some examples
  14. Some proofs
  15. Proof of Theorem \ref{['optmin']}
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $f = f_{1}+\dots+f_m$ be such that each $f_i\in\mathbb{R}[x_{\Delta_{i}}]$ and let $G = G_1 + \cdots + G_m$ be such that each $G_i \subseteq \mathbb{R}[x_{\Delta_{i}}]$. Then, for a given $\gamma\in\mathbb{R}$, it holds $f-\gamma \in G$ if and only if there exist polynomials $p_i \in \mathbb{R}[

Theorems & Definitions (38)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Example 3.4
  • ...and 28 more