Scaled Relative Graph: Nonexpansive operators via 2D Euclidean Geometry

Ryu, Ernest K.; Hannah, Robert; Yin, Wotao

doi:10.48550/arxiv.1902.09788

Cited by 6 publications

(21 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following two propositions demonstrate the verification of system properties from the system's SRG, and follow directly from [1,Prop. 3.3 & Thm.…”

Section: Scaled Relative Graphsmentioning

confidence: 98%

“…We define SRGs in the same way as Ryu, Hannah, and Yin [1], with the minor modification of allowing complex valued inner products. Let H be a Hilbert space, equipped with an inner product, ⟨⋅ ⋅⟩ ∶ H×H → C, and the induced norm…”

Section: Scaled Relative Graphsmentioning

confidence: 99%

“…The properties of bounded incremental L 2 gain and incremental positivity are particular examples of incremental Integral Quadratic Constraints (IQCs) [13]. A striking corollary of Ryu, Hannah, and Yin [1,Thm. 3.5] is that any SRG defined by an incremental IQC is SRG-full.…”

Section: Scaled Relative Graphsmentioning

confidence: 99%

“…The Scaled Relative Graph (SRG) of Ryu, Hannah, and Yin [1] allows the action of a nonlinear operator to be visualized on the complex plane. Incremental properties of an operator, measured between pairs of inputs and outputs, such as maximal monotonicity and Lipschitz continuity, may be verified by checking geometric conditions on the SRG of the operator.…”

Section: Introductionmentioning

confidence: 99%

“…This tool allows simple, intuitive and rigorous proofs of the convergence of many algorithms in convex optimization. Furthermore, the graphical method is particularly suitable for proving tightness of convergence bounds, with several novel tightness results being proved [1], [2].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Scaled relative graphs for system analysis

Chaffey

Forni

Sepulchre

2021

2021 60th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on L 2 is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function and quadratic constraints, allowing system properties to be verified or disproved. Interconnections of systems correspond to graphical manipulations of their SRGs. This is used to provide a simple, graphical proof of the classical incremental passivity theorem.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.

show abstract

“…The following two propositions demonstrate the verification of system properties from the system's SRG, and follow directly from [1,Prop. 3.3 & Thm.…”

Section: Scaled Relative Graphsmentioning

confidence: 98%

Section: Scaled Relative Graphsmentioning

confidence: 99%

Section: Scaled Relative Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Scaled relative graphs for system analysis

Chaffey

Forni

Sepulchre

2021

2021 60th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

show abstract

Finding the Forward-Douglas–Rachford-Forward Method

Ryu

Vũ

2019

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

We consider the monotone inclusion problem with a sum of 3 operators, in which 2 are monotone and 1 is monotone-Lipschitz. The classical Douglas-Rachford and Forward-backward-forward methods respectively solve the monotone inclusion problem with a sum of 2 monotone operators and a sum of 1 monotone and 1 monotone-Lipschitz operators. We first present a method that naturally combines Douglas-Rachford and Forward-backwardforward and show that it solves the 3 operator problem under further assumptions, but fails in general. We then present a method that naturally combines Douglas-Rachford and forward-reflected-backward, a recently proposed alternative to Forward-backward-forward by Malitsky and Tam [arXiv:1808.04162, 2018]. We show that this second method solves the 3 operator problem generally, without further assumptions. IntroductionWe consider the monotone inclusion problem of finding a zero of the sum of 2 maximal monotone and 1 monotone-Lipschitz operators. The classical Communicated by Jalal Fadili. Recently, there has been much work developing splitting methods combining and unifying classical ones. Another classical method is forward-backward (FB) splitting [3,4], which solves the problem with a sum of 1 monotone and 1 cocoercive operators. The effort of combining DR and FB was started by Raguet, Fadili, and Peyré [5,6], extended by Briceño-Arias [7], and completed by Davis and Yin [8] as they proved convergence for the sum of 2 monotone and 1 cocoercive operators. FB and FBF were combined by Briceño-Arias and Davis [9] as they proved convergence for 1 monotone, 1 cocoercive, and 1 monotone-Lipschitz operators. These combined splitting methods can efficiently solve monotone inclusion problems with more complex structure.On the other hand, DR and FBF have not been fully combined, to the best of our knowledge. Banert's relaxed forward backward (in the thesis [10]) and Briceño-Arias's forward-partial inverse-forward [11] combine DR and FBF in the setup where one operator is a normal cone operator with respect to a closed subspace. However, neither method applies to the general setup with 2 maximal monotone and 1 monotone-Lipschitz operators.In this work, we first present a method that naturally combines and unifies DR and FBF. We prove convergence under further assumptions, and we prove, through a counterexample, that convergence cannot be established in full generality. We then propose a second method that naturally combines and unifies DR and forward-reflected-backward (FRB), a recently proposed alternative to FBF by Malitsky and Tam [12]. We show that this combination of DR and FRB does converge in full generality.The paper is organized as follows. Section 2 states the problem formally. Section 3 reviews preliminary information and sets up the notation. Section 4 presents our first proposed method combining DR and FBF, proves convergence under certain further assumptions, and proves divergence in the fully general case. Section 5 presents our second proposed method combining DR and FRB and proves convergen...

show abstract