Exploiting Term Sparsity in Moment-SOS hierarchy for Dynamical Systems

Abstract

In this paper we present term sparsity sum-of-squares (TSSOS) methods applied to several problems from dynamical systems, such as region of attraction, maximum positively invariant sets and global attractors. We combine the TSSOS algorithm of Wang, Magron and Lasserre [SIAM J. Optim., 31(1):30-58, 2021] with existing infinite dimensional linear program representations of those sets. This leads to iterative schemes in the moment-sumof-squares hierarchy which allows less expensive computations while keeping convergence guarantees. Finally this procedure is closely related to sign symmetries of the dynamical system as was already revealed for polynomial optimization. Numerical examples demonstrate the efficiency of the approach in the presence of appropriate sparsity.

Converging outer approximations to global attractors using semidefinite programming

“…The cost of having guaranteed bounds and finding the global optima using SOS methods comes at the price of expensive scaling, that is, high degree d or dimension n makes the SDP intracktable for current solvers/memory/computation power. Therefore, further structure should be exploited, such as sparsity [20], [22], [24] or symmetry [8].…”