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Exploiting Term Sparsity in Symmetry-Adapted Basis for Polynomial Optimization

Igor Klep, Victor Magron, Tobias Metzlaff, Jie Wang

TL;DR

The paper tackles nonconvex polynomial optimization with finite-group invariance by augmenting the Lasserre moment-SOS hierarchy with a symmetry-adapted basis and term sparsity. It shows that reformulating in a symmetry-aware basis yields block-diagonal SDP relaxations without changing the bounds, and extends term-sparsity techniques to operate inside each symmetry block via a generalized TSSOS construction. Convergence guarantees under the Archimedean property and maximal chordal extensions accompany practical demonstrations on dihedral, product cyclic, and symmetric quartics, where the combined approach outperforms dense and prior sparse methods. An accompanying Julia implementation (TSSOS) and discussions on orbit-space reductions and equivariant dynamical systems highlight the method’s scalability and potential for broad applicability in invariant polynomial optimization and related dynamical systems analysis.

Abstract

Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function and constraint polynomials are invariant under the action of a finite group. The present paper simultaneously exploits group symmetry and term sparsity in order to reduce the computational cost of the hierarchy. We first exploit symmetry by writing the semidefinite matrices in a symmetry-adapted basis according to an isotypic decomposition. The matrices in such a basis are block diagonal. Secondly, we exploit term sparsity on each block to further reduce the optimization matrix variables. This is a non-trivial extension of the term sparsity-based hierarchy related to sign symmetry that was introduced by two of the authors. Our method is compared with existing techniques via benchmarks on quartics with dihedral, cyclic and symmetric group symmetry.

Exploiting Term Sparsity in Symmetry-Adapted Basis for Polynomial Optimization

TL;DR

The paper tackles nonconvex polynomial optimization with finite-group invariance by augmenting the Lasserre moment-SOS hierarchy with a symmetry-adapted basis and term sparsity. It shows that reformulating in a symmetry-aware basis yields block-diagonal SDP relaxations without changing the bounds, and extends term-sparsity techniques to operate inside each symmetry block via a generalized TSSOS construction. Convergence guarantees under the Archimedean property and maximal chordal extensions accompany practical demonstrations on dihedral, product cyclic, and symmetric quartics, where the combined approach outperforms dense and prior sparse methods. An accompanying Julia implementation (TSSOS) and discussions on orbit-space reductions and equivariant dynamical systems highlight the method’s scalability and potential for broad applicability in invariant polynomial optimization and related dynamical systems analysis.

Abstract

Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function and constraint polynomials are invariant under the action of a finite group. The present paper simultaneously exploits group symmetry and term sparsity in order to reduce the computational cost of the hierarchy. We first exploit symmetry by writing the semidefinite matrices in a symmetry-adapted basis according to an isotypic decomposition. The matrices in such a basis are block diagonal. Secondly, we exploit term sparsity on each block to further reduce the optimization matrix variables. This is a non-trivial extension of the term sparsity-based hierarchy related to sign symmetry that was introduced by two of the authors. Our method is compared with existing techniques via benchmarks on quartics with dihedral, cyclic and symmetric group symmetry.

Paper Structure

This paper contains 21 sections, 13 theorems, 69 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary Background on Symmetric Sums of Squares
  3. Moment Relaxation and SOS Reinforcement
  4. Linear Representation Theory
  5. Symmetry Reduction via Symmetry-Adapted Bases
  6. Symmetry Exploitation for Sparse Problems
  7. Chordal Graphs and Sparse Matrices
  8. Term Sparsity
  9. Example
  10. Generalized TSSOS Hierarchy and Asymptotic Convergence
  11. Benchmarks
  12. 1-D Ring Ising Quartic with Dihedral Symmetry
  13. 2-D Torus Grid Quartic with Product Cyclic Symmetry
  14. Symmetric Quartic
  15. Discussion of the Numerical Experiments
  16. ...and 6 more sections

Key Result

Theorem 2.1

putinar93 Assume that $\mathrm{QM}(\mathbf{g})$ is Archimedean.

Figures (3)

  • Figure 1: Maximal SDP block sizes, number of SDP constraints and total computation time versus $(n,\,r)$ for the $1$-D ring Ising quartic with dihedral symmetry.
  • Figure 2: Maximal SDP block sizes, number of SDP constraints and total computation time versus $(p,\,q,\,r,\,s)$ for the $2$-D torus grid quartic with product cyclic symmetry.
  • Figure 3: Maximal SDP block sizes versus $(n,\,r,\,s)$ for the symmetric quartic.

Theorems & Definitions (29)

  • Theorem 2.1: Putinar's Positivstellensatz
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Symmetry-Adapted Basis
  • Definition 2.5
  • Lemma 2.6
  • Corollary 2.7: Block Diagonalization
  • Theorem 2.8: Symmetric Positivstellensatz
  • Theorem 2.9
  • Corollary 2.10
  • ...and 19 more