Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

Zheng, Yang; Fantuzzi, Giovanni; Papachristodoulou, Antonis

doi:10.1016/j.arcontrol.2021.09.001

Cited by 30 publications

(21 citation statements)

References 153 publications

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“…In another line of research, techniques based on properties and algorithms for chordal sparsity patterns have been applied to semidefinite programming since the late 1990s [3,13,18,29,30,34,35,42,46,50,51,58]; see [54,60] for recent surveys. An important tool from this literature is the conversion or clique decomposition method proposed by Fukuda et al [30,42].…”

Section: Introductionmentioning

confidence: 99%

Bregman primal–dual first-order method and application to sparse semidefinite programming

Xin¹,

Vandenberghe²

2021

Comput Optim Appl

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We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

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Section: Introductionmentioning

confidence: 99%

Bregman primal–dual first-order method and application to sparse semidefinite programming

Xin¹,

Vandenberghe²

2021

Comput Optim Appl

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“…The first approach, based on semidefinite programming, is extremely flexible, has general convergence guarantees, and can be implemented using general-purpose interior-point SDP solvers that are available open-source. To use such solvers in turbulent regimes that require accurate discretization of thin boundary layers, however, one must employ advanced decomposition techniques that have only recently been developed by the optimization community (see [97] for a review). The second approach to optimizing background fields we have reviewed, instead, relies on timestepping.…”

Section: Discussionmentioning

confidence: 99%

“…The structure of each block can be exploited in a similar way using chordal decomposition techniques for SDPs [93][94][95][96][97], which decompose sparse LMIs into smaller ones by considering their dense principal submatrices (see the bottom panel in figure 2 for an illustration). This requires introducing additional optimization variables to account for the overlap between dense submatrices, but, if these are small and do not overlap significantly, then the added cost is negligible compared to the savings associated by the reduction in LMI dimension.…”

Section: B)mentioning

confidence: 99%

The background method: theory and computations

Fantuzzi

Arslan

Wynn

2022

Phil. Trans. R. Soc. A.

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The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader ‘auxiliary function’ framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh–Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.

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“…Still, SOS programs suffer from some limitations that have been or are being addressed in the literature. A first one is scalability of SOS programs: tools are available [24] that automatically reduce the problem size without major computational costs, and recent works on large-scale semidefinite and polynomial optimization [41] improve scalability of SOS programs significantly. Secondly, it is often the case that the obtained SOS program is bilinear in the decision variables.…”

Section: In the Control Community Invariance For Linear Systemsmentioning

confidence: 99%

Data-driven design of safe control for polynomial systems

Luppi¹,

Bisoffi²,

Persis³

et al. 2021

Preprint

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We consider the problem of designing an invariant set using only a finite set of input-state data collected from an unknown polynomial system in continuous time. We consider noisy data, i.e., corrupted by an unknown-but-bounded disturbance. We derive a data-dependent sum-of-squares program that enforces invariance of a set and also optimizes the size of the invariant set while keeping it within a set of user-defined safety constraints; the solution of this program directly provides a polynomial invariant set and a state-feedback controller. We numerically test the design on a system of two platooning cars.

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Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization

Cited by 30 publications

References 153 publications

Bregman primal–dual first-order method and application to sparse semidefinite programming

Bregman primal–dual first-order method and application to sparse semidefinite programming

The background method: theory and computations

Data-driven design of safe control for polynomial systems

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