[Paper Review] Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations

Corbinian Schlosser, Milan Korda|arXiv (Cornell University)|Dec 10, 2020

Advanced Optimization Algorithms Research7 citations

TL;DR

This paper introduces a sparse decomposition framework for nonlinear dynamical systems based on causal dependence structures, enabling dimension reduction in computing key sets like the region of attraction, maximum positively invariant set, and global attractor. By exploiting subsystem sparsity, it enables convergence-preserving, low-dimensional sum-of-squares relaxations via convex optimization, significantly reducing semidefinite program size without sacrificing accuracy.

ABSTRACT

In this paper, we propose a general sparse decomposition of dynamical systems provided that the vector field and constraint set possess certain sparse structures, which we call subsystems. This notion is based on causal dependence in the dynamics between the different states. This results in sparse descriptions for fundamental problems from nonlinear dynamical systems: region of attraction, maximum positively invariant set, and global attractor. The decompositions can be paired with any method for computing (outer) approximations of these sets to reduce the computation to lower dimensional systems. This is illustrated by methods from previous work based on infinite-dimensional linear programming. This exhibits one example where the curse of dimensionality is present and hence dimension reduction is crucial. In this context, for polynomial dynamics, we show that these problems admit a sparse sum-of-squares (SOS) approximation with guaranteed convergence such that the number of variables in the largest SOS multiplier is given by the dimension of the largest subsystem appearing in the decomposition. The dimension of such subsystems depends on the sparse structure of the vector field and the constraint set; if the dimension of the largest subsystem is small compared to the ambient dimension, this allows for a significant reduction in the computation time of the SOS approximations. Numerical examples accompany the approach.

Motivation & Objective

• To address the curse of dimensionality in computing fundamental sets in nonlinear dynamical systems, such as the region of attraction (ROA), maximum positively invariant (MPI) set, and global attractor (GA).
• To develop a general framework that decomposes high-dimensional systems into causally independent subsystems to reduce computational complexity.
• To preserve convergence guarantees when applying moment-sum-of-squares (SOS) relaxations to these sets, even under sparsity exploitation.
• To provide a coordinate-free, structure-aware method applicable to any outer approximation method with convergence properties, extending beyond specific solvers.

Proposed method

• The method identifies subsystems based on causal dependence in the vector field and constraint set, defining partitions of state variables where dynamics are causally decoupled.
• It introduces a graph-based structure where each subsystem corresponds to a minimal partition inducing a factorization of the state space, ensuring that dynamics in one subsystem do not causally affect others.
• The framework leverages the running intersection property and sparsity in polynomial dynamics to construct sparse sum-of-squares (SOS) relaxations with the largest SOS multiplier dimension equal to the largest subsystem size.
• It applies the moment-sum-of-squares hierarchy to each subsystem independently, preserving convergence to the true sets from the outside.
• The approach is general and compatible with any outer approximation method (e.g., infinite-dimensional LP, set-oriented methods) that satisfies convergence guarantees.
• The decomposition is computed via a topology on index sets, where minimal elements of a closed family of partitions define the finest factorization of the state space.

Experimental results

Research questions

• RQ1 Can causal sparsity in nonlinear dynamical systems be exploited to decompose high-dimensional problems into lower-dimensional subsystems without losing convergence guarantees?
• RQ2 How can sparse sum-of-squares relaxations be constructed for the region of attraction, MPI set, and global attractor while preserving convergence?
• RQ3 What structural conditions on the vector field and constraint set allow for a valid and minimal subsystem decomposition?
• RQ4 Can the moment-sum-of-squares hierarchy be applied to subsystems independently while maintaining convergence to the true invariant sets?
• RQ5 Is it possible to achieve dimension reduction in semidefinite programming relaxations for dynamical systems using a coordinate-free, structure-aware decomposition?

Key findings

• The method enables convergence-preserving, low-dimensional sum-of-squares relaxations for the region of attraction, MPI set, and global attractor by decomposing the system into causally independent subsystems.
• The largest semidefinite program in the SOS relaxation depends only on the size of the largest subsystem, not the full system, enabling significant computational savings.
• The framework is general and can be applied to any outer approximation method with convergence guarantees, including the moment-sum-of-squares hierarchy and set-oriented methods.
• The decomposition is based on a minimal partition of state variables derived from a closed family of index sets, ensuring the finest possible factorization of the state space.
• The approach avoids the loss of convergence seen in prior sparsity methods by exploiting structural sparsity in the dynamics rather than algebraic sparsity in polynomials.
• Numerical examples demonstrate the effectiveness of the method in reducing SDP size while maintaining convergence to the true invariant sets.