Approximating regions of attraction of a sparse polynomial differential system

Tacchi, Matteo; Cardozo, Carmen; Henrion, Didier; Lasserre, Jean-Bernard

doi:10.1016/j.ifacol.2020.12.1488

Cited by 25 publications

(21 citation statements)

References 17 publications

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“…Many applications of interest have been successfully handled thanks to this framework, for instance certified roundoff error bounds in computer arithmetics [MCD17,Mag18] with up to several hundred variables and constraints, optimal powerflow problems [JM18] with up to several thousands of variables and constraints. More recent extensions have been developed for volume computation of sparse semialgebraic sets [TWLH19], approximating regions of attraction of sparse polynomial systems [TCHL19], noncommutative POPs [KMP19], Lipschitz constant estimation of deep networks [LRC20, CLMP20] and for sparse positive definite functions [MML20]. In these applications both polynomial cost functions and polynomial constraints have a specific correlative sparsity pattern.…”

Section: Related Work For Unconstrained Popsmentioning

confidence: 99%

CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization

Wang,

Magron,

Lasserre

et al. 2020

Preprint

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This work proposes a new moment-SOS hierarchy, called CS-TSSOS, for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [WKKM06] while the latter was initially proposed by Wang et al. [WLX19] and later exploited in the TSSOS hierarchy [WML19, WML20].In doing so we obtain CS-TSSOS -a two-level hierarchy of semidefinite programming relaxations with (i), the crucial property to involve quasi block-diagonal matrices and (ii), the guarantee of convergence to the global optimum. We demonstrate its efficiency on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to several thousands of variables and ten thousands of constraints.

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Section: Related Work For Unconstrained Popsmentioning

confidence: 99%

CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization

Wang,

Magron,

Lasserre

et al. 2020

Preprint

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“…As a result, the largest polynomial degree that can currently be considered for the nine-dimensional system in subsection 4.3 is approximately 10 on a workstation with 64GB of RAM, and reduces to no more than 4 or 6 for ODEs with a few tens of states. Nevertheless, removing computational bottlenecks in general polynomial optimization and in its applications to dynamical systems are problems that have attracted significant interest in recent years (see, e.g., [28][29][30][31][32][33][34][35][36][37]), so we expect that our UPO search strategy will become practical for ODEs of moderate dimension in the near future. With computational aspects in mind, a particularly attractive aspect of the control strategy proposed in this paper is that its four steps do not depend on the particular algorithms used to carry them out.…”

Section: Discussionmentioning

confidence: 99%

Finding unstable periodic orbits: a hybrid approach with polynomial optimization

Lakshmi,

Fantuzzi,

Chernyshenko

et al. 2021

Preprint

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We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems parameterized by a scalar k such that the original ODE system is recovered for k = 0, and such that the optimal orbit is less unstable, or even stabilized, for k > 0. Periodic orbits for the controlled system can be more easily converged with traditional methods and numerical continuation in k allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics.

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“…A possible approach to tackle this problem is to exploit symmetries or sparsity of the problem. While symmetry exploitation comes at no cost of accuracy [31], obtaining a lossless (or at least convergence-preserving) sparse relaxation in this dynamical context is currently an open challenge (see [30] for results in this direction). Although there are further interesting topological properties to the global attractor some of them are invisible to our approach due to the fact that the primal linear program we use can only identify the global attractor up to a set of Lebesgue measure zero which excludes that we have control of certain topological properties.…”

Section: Discussionmentioning

confidence: 99%

Converging outer approximations to global attractors using semidefinite programming

Schlosser¹,

Korda²

2020

Preprint

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This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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Approximating regions of attraction of a sparse polynomial differential system

Cited by 25 publications

References 17 publications

CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization

CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization

Finding unstable periodic orbits: a hybrid approach with polynomial optimization

Converging outer approximations to global attractors using semidefinite programming

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