Robust and structure exploiting optimisation algorithms: an integral quadratic constraint approach

Michalowsky, Simon; Scherer, Carsten W.; Ebenbauer, Christian

doi:10.1080/00207179.2020.1745286

Cited by 41 publications

(41 citation statements)

References 40 publications

(147 reference statements)

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“…Remark 3: Theorem 1 and 2 are sufficient for asymptotic stability. Furthermore, in recent studies [22], [23], a subset of FIR Zames-Falb multipliers are invoked to explore the exponential decay rate of Lurye systems for which asymptotic stability is guaranteed. Similarly, if the Lyapunov function is further bounded by α∥x i ∥ p ≤ V (x i ) ≤ β ∥x i ∥ p and certifies ρV (x i+1 ) −V (x i ) ≤ 0 with some α, β > 0, p ≥ 1 and ρ > 1, then the system is proved to be exponentially stable in the way ∥x i ∥ ≤ (β /α) 1/p ∥x 0 ∥(ρ 1/p ) −i [21, Theorem 13.2].…”

Section: B Frequency Domain Interpretationmentioning

confidence: 99%

“…Similarly, if the Lyapunov function is further bounded by α∥x i ∥ p ≤ V (x i ) ≤ β ∥x i ∥ p and certifies ρV (x i+1 ) −V (x i ) ≤ 0 with some α, β > 0, p ≥ 1 and ρ > 1, then the system is proved to be exponentially stable in the way ∥x i ∥ ≤ (β /α) 1/p ∥x 0 ∥(ρ 1/p ) −i [21, Theorem 13.2]. Then, it is also possible to show the equivalence between the Lyapunov criterion and the Zames-Falb theorem in [22], [23] for exponential stability, but this problem is still open.…”

Section: B Frequency Domain Interpretationmentioning

confidence: 99%

See 1 more Smart Citation

A Lyapunov-Lurye Functional Parametrization of Discrete-Time Zames-Falb Multipliers

Zhang

Carrasco

2022

IEEE Control Syst. Lett.

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We consider the absolute stability of discrete-time Lurye systems with SISO/MIMO (non-repeated SISO) nonlinearities that are sector bounded and slope restricted. For this class of systems, we present a parametrization of Lyapunov-Lurye functional (LLF) that is the time-domain equivalence to finite impulse response (FIR) Zames-Falb multipliers. As searches over FIR Zames-Falb multipliers provide the bestknown results in the literature, the parametrization here provides the best-known Lyapunov function for absolute stability. A motivation of this alternative is making it easy to analyze the system in the time domain, especially when the frequency domain expression of the system is not straightforward. In this letter, we show the equivalence between the proposed LLF and FIR multipliers theoretically and numerically.

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Section: B Frequency Domain Interpretationmentioning

confidence: 99%

Section: B Frequency Domain Interpretationmentioning

confidence: 99%

A Lyapunov-Lurye Functional Parametrization of Discrete-Time Zames-Falb Multipliers

Zhang

Carrasco

2022

IEEE Control Syst. Lett.

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“…where the inequality is obtained by applying (18) with M =Â i for each i. Now, substitution for a i and b i into the above inequality yields…”

Section: B Eliminating the Transient Growth Using Balanced Initial Conditions (Theorem 3)mentioning

confidence: 99%

Robustness of Accelerated First-Order Algorithms for Strongly Convex Optimization Problems

Mohammadi

Razaviyayn

Jovanović

2021

IEEE Trans. Automat. Contr.

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Optimization algorithms are increasingly being used in applications with limited time budgets. In many real-time and embedded scenarios, only a few iterations can be performed and traditional convergence metrics cannot be used to evaluate performance in these non-asymptotic regimes. In this paper, we examine the transient behavior of accelerated first-order optimization algorithms. For quadratic optimization problems, we employ tools from linear systems theory to show that transient growth arises from the presence of non-normal dynamics. We identify the existence of modes that yield an algebraic growth in early iterations and quantify the transient excursion from the optimal solution caused by these modes. For strongly convex smooth optimization problems, we utilize the theory of integral quadratic constraints to establish an upper bound on the magnitude of the transient response of Nesterov's accelerated method. We show that both the Euclidean distance between the optimization variable and the global minimizer and the rise time to the transient peak are proportional to the square root of the condition number of the problem. Finally, for problems with large condition numbers, we demonstrate tightness of the bounds that we derive up to constant factors.

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“…Convex searches leading to numerical criteria have been proposed in [2,7,25], and it is worth highlighting its role in convergence analysis of optimisation algorithms, e.g. [8,17,18,21,32]. The efficiency of the searches for discrete-time OZF multipliers has generated questions on the conservatism of the sufficient condition with OZF multipliers, e.g.…”

Section: Introductionmentioning

confidence: 99%

On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

Su¹,

Seiler²,

Carrasco³

et al. 2021

Preprint

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This paper considers the robust stability of a discrete-time Lurye system consisting of the feedback interconnection between a linear system and a bounded and monotone nonlinearity. It has been conjectured that the existence of a suitable linear time-invariant (LTI) O'Shea-Zames-Falb multiplier is not only sufficient but also necessary. Roughly speaking, a successful proof of the conjecture would require: (a) a conic parameterization of a set of multipliers that describes exactly the set of nonlinearities, (b) a lossless S-procedure to show that the non-existence of a multiplier implies that the Lurye system is not uniformly robustly stable over the set of nonlinearities, and (c) the existence of a multiplier in the set of multipliers used in (a) implies the existence of an LTI multiplier. We investigate these three steps, showing the current bottlenecks for proving this conjecture. In addition, we provide an extension of the class of multipliers which may be used to disprove the conjecture.

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Robust and structure exploiting optimisation algorithms: an integral quadratic constraint approach

Cited by 41 publications

References 40 publications

A Lyapunov-Lurye Functional Parametrization of Discrete-Time Zames-Falb Multipliers

A Lyapunov-Lurye Functional Parametrization of Discrete-Time Zames-Falb Multipliers

Robustness of Accelerated First-Order Algorithms for Strongly Convex Optimization Problems

On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

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