Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparse Polynomial Optimization with Unbounded Sets

Lei Huang, Shucheng Kang, Jie Wang, Heng Yang

TL;DR

The proposed sparse homogenized Moment-SOS hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern to solve sparse polynomial optimization problems on unbounded sets with up to thousands of variables.

Abstract

This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy has asymptotic convergence. Furthermore, we provide two alternative sparse hierarchies to remove perturbations while preserving asymptotic convergence. As byproducts, new Positivstellensätze are obtained for sparse positive polynomials on unbounded sets. Extensive numerical experiments demonstrate the power of our approach in solving sparse polynomial optimization problems on unbounded sets with up to thousands of variables. Finally, we apply our approach to tackle two trajectory optimization problems (block-moving with minimum work and optimal control of Van der Pol).

Sparse Polynomial Optimization with Unbounded Sets

TL;DR

The proposed sparse homogenized Moment-SOS hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern to solve sparse polynomial optimization problems on unbounded sets with up to thousands of variables.

Abstract

This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy has asymptotic convergence. Furthermore, we provide two alternative sparse hierarchies to remove perturbations while preserving asymptotic convergence. As byproducts, new Positivstellensätze are obtained for sparse positive polynomials on unbounded sets. Extensive numerical experiments demonstrate the power of our approach in solving sparse polynomial optimization problems on unbounded sets with up to thousands of variables. Finally, we apply our approach to tackle two trajectory optimization problems (block-moving with minimum work and optimal control of Van der Pol).
Paper Structure (20 sections, 12 theorems, 89 equations, 2 figures, 9 tables)

This paper contains 20 sections, 12 theorems, 89 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Some basics for polynomial optimization
  4. The homogenized Moment-SOS hierarchy
  5. Polynomial optimization with correlative sparsity
  6. The sparse homogenized Moment-SOS hierarchy with perturbations
  7. Convergence analysis
  8. Sparse Positivstellensätze with perturbations
  9. Extraction of minimizers
  10. Sparse homogenized Moment-SOS hierarchy without perturbations
  11. Convergence analysis
  12. Sparse Positivstellensätze without perturbations
  13. An alternative sparse homogenized Moment-SOS hierarchy without perturbations
  14. Numerical examples
  15. Unconstrained polynomial optimization
  16. ...and 5 more sections

Key Result

Theorem 2.1

\newlabelspaput0 Suppose that (1.1) has the csp $(\mathbf{x}(1),\dots,\mathbf{x}(p))$ satisfying the RIP, and the quadratic module $\hbox{QM}[(g_j)_{j\in J_{\ell}},\mathbf{x}(\ell)]$ is Archimedean for each $\ell \in [p]$. If $f>0$ on $K$, then

Figures (2)

  • Figure 1: Comparison between SDP's solutions (blue lines) and solutions refined by fmincon (red lines) in the block-moving example. In HSSOS3, red lines and blue lines are nearly indistinguishable, indicating the attainment of tight solutions.
  • Figure 2: Comparison between SDP's solutions and solutions refined by fmincon in the Van der Pol example. Notably, all three algorithms' initial guesses lead to the same refined trajectories. Among these initial guesses, the one offered by HSSOS3 is of the best quality.

Theorems & Definitions (31)

  • Theorem 2.1: grimm2007notekojima2009notelasserre2006convergent
  • Remark 2.2
  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Lemma 3.3
  • Proof 2
  • Theorem 3.4
  • Proof 3
  • Remark 3.5
  • ...and 21 more