Sparse semidefinite programs with guaranteed near-linear time complexity via dualized clique tree conversion

Zhang, Richard Y.; Lavaei, Javad

doi:10.1007/s10107-020-01516-y

Cited by 19 publications

(12 citation statements)

References 64 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%

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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%

“…This equivalent problem may be considerably easier to solve by interior-point methods than the original SDP (1). Recent examples where the clique decomposition is applied to solve large sparse SDPs can be found in [27,58].…”

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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“…However, SDP relaxation can be computationally demanding and may not scale well to large-scale matrices [40].…”

Section: B Convexificationmentioning

confidence: 99%

Optimal Multi-Robot Motion Planning via Parabolic Relaxation

Choi¹,

Adil²,

Rahmani³

et al. 2021

Preprint

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Multi-robot systems offer enhanced capability over their monolithic counterparts, but they come at a cost of increased complexity in coordination. To reduce complexity and to make the problem tractable, multi-robot motion planning (MRMP) methods in the literature adopt de-coupled approaches that sacrifice either optimality or dynamic feasibility. In this paper, we present a convexification method, namely "parabolic relaxation", to generate optimal and dynamically feasible trajectories for MRMP in the coupled joint-space of all robots. We leverage upon the proposed relaxation to tackle the problem complexity and to attain computational tractability for planning over one hundred robots in extremely clustered environments. We take a multistage optimization approach that consists of i) mathematically formulating MRMP as a non-convex optimization, ii) lifting the problem into a higher dimensional space, iii) convexifying the problem through the proposed computationally efficient parabolic relaxation, and iv) penalizing with iterative search to ensure feasibility and recovery of feasible and near-optimal solutions to the original problem. Our numerical experiments demonstrate that the proposed approach is capable of generating optimal and dynamically feasible trajectories for challenging motion planning problems with higher success rate than the state-of-the-art, yet remain computationally tractable for over one hundred robots in a highly dense environment.

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“…This section reviews two general approaches for doing so. The first one, similar to the conversion methods in Section 3.3.1, reformulates problems (3.5) and (3.6) as SDPs with small positive semidefinite cones, which are often easier to solve with general-purpose interior-point solvers (Fukuda et al, 2001;Kim et al, 2011;Nakata et al, 2003;Zhang & Lavaei, 2020b). The second approach, instead, directly solves (3.6)-(3.5) using an interior-point method for nonsymmetric conic optimization (Andersen et al, 2010a;Coey et al, 2020;Nesterov, 2012;Skajaa & Ye, 2015).…”

Section: Interior-point Algorithmsmentioning

confidence: 99%

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

Zheng,

Fantuzzi,

Papachristodoulou

2021

Preprint

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Chordal and factor-width decomposition methods for semidefinite programming and polynomial optimization have recently enabled the analysis and control of large-scale linear systems and medium-scale nonlinear systems. Chordal decomposition exploits the sparsity of semidefinite matrices in a semidefinite program (SDP), in order to formulate an equivalent SDP with smaller semidefinite constraints that can be solved more efficiently. Factor-width decompositions, instead, relax or strengthen SDPs with dense semidefinite matrices into more tractable problems, trading feasibility or optimality for lower computational complexity. This article reviews recent advances in large-scale semidefinite and polynomial optimization enabled by these two types of decomposition, highlighting connections and differences between them. We also demonstrate that chordal and factor-width decompositions allow for significant computational savings on a range of classical problems from control theory, and on more recent problems from machine learning. Finally, we outline possible directions for future research that have the potential to facilitate the efficient optimization-based study of increasingly complex large-scale dynamical systems.

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Sparse semidefinite programs with guaranteed near-linear time complexity via dualized clique tree conversion

Cited by 19 publications

References 64 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

Optimal Multi-Robot Motion Planning via Parabolic Relaxation

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

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