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Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension

Jie Wang, Victor Magron, Jean-Bernard Lasserre

TL;DR

The chordal-TSSOS hierarchy that is proposed is a new sparse moment-SOS framework based on term-sparsity and chordal extension, which is a two-level hierarchy of semidefinite programming relaxations for solving polynomial optimization problems (POPs).

Abstract

This work is a follow-up and a complement to arXiv:1912.08899 [math.OC] for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose is a new sparse moment-SOS framework based on term-sparsity and chordal extension. By exploiting term-sparsity of the input polynomials we obtain a two-level hierarchy of semidefinite programming relaxations. The novelty and distinguishing feature of such relaxations is to obtain quasi block-diagonal matrices obtained in an iterative procedure that performs chordal extension of certain adjacency graphs. The graphs are related to the terms arising in the original data and not to the links between variables. Various numerical examples demonstrate the efficiency and the scalability of this new hierarchy for both unconstrained and constrained POPs. The two hierarchies are complementary. While the former TSSOS arXiv:1912.08899 [math.OC] has a theoretical convergence guarantee, the chordal-TSSOS has superior performance but lacks this theoretical guarantee.

Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension

TL;DR

The chordal-TSSOS hierarchy that is proposed is a new sparse moment-SOS framework based on term-sparsity and chordal extension, which is a two-level hierarchy of semidefinite programming relaxations for solving polynomial optimization problems (POPs).

Abstract

This work is a follow-up and a complement to arXiv:1912.08899 [math.OC] for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose is a new sparse moment-SOS framework based on term-sparsity and chordal extension. By exploiting term-sparsity of the input polynomials we obtain a two-level hierarchy of semidefinite programming relaxations. The novelty and distinguishing feature of such relaxations is to obtain quasi block-diagonal matrices obtained in an iterative procedure that performs chordal extension of certain adjacency graphs. The graphs are related to the terms arising in the original data and not to the links between variables. Various numerical examples demonstrate the efficiency and the scalability of this new hierarchy for both unconstrained and constrained POPs. The two hierarchies are complementary. While the former TSSOS arXiv:1912.08899 [math.OC] has a theoretical convergence guarantee, the chordal-TSSOS has superior performance but lacks this theoretical guarantee.

Paper Structure

This paper contains 12 sections, 12 theorems, 68 equations, 5 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. SOS polynomials
  4. Moment-SOS relaxations for POPs
  5. Chordal graphs and sparse matrices
  6. The chordal-TSSOS Hierarchy: Unconstrained case
  7. The chordal-TSSOS Hierarchy: Constrained case
  8. Computational cost discussion
  9. Numerical experiments
  10. Unconstrained polynomial optimization problems
  11. Constrained polynomial optimization problems
  12. Conclusions and outlooks

Key Result

Theorem 2.1

Let $G(V,E)$ be a chordal graph and assume that $C_1,\ldots,C_t$ are all of the maximal cliques of $G(V,E)$. Then a matrix $Q\in\mathbf{S}_+^{|V|}\cap\mathbf{S}_G$ if and only if there exist $Q_{k}\in \mathbf{S}_+^{|C_k|}$ for $k=1,\ldots,t$ such that $Q=\sum_{k=1}^tP_{C_k}^TQ_{k}P_{C_k}$.

Figures (5)

  • Figure 1: The support-extension $\hbox{\rm{SE}}(G)$ of $G$
  • Figure 2: The chordal-extension $\overline{G}$ of $G$
  • Figure 3: The tsp graph $G_0$ and its chordal-extension $G_1$ for Example \ref{['ex1']}
  • Figure 4: The tsp graph $G_0$ of $f$
  • Figure 5: The tsp graph $G_0$ of $f_n$

Theorems & Definitions (34)

  • Theorem 2.1: va, Theorem 9.2
  • Theorem 2.2: va, Theorem 10.1
  • Example 3.1
  • Example 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Example 3.5
  • Remark 3.6
  • Proposition 3.7
  • ...and 24 more