Invariance and efficiency of convex representations

Chua, Chek Beng; Tunçel, Levent

doi:10.1007/s10107-006-0072-6

Cited by 22 publications

(13 citation statements)

References 21 publications

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“…Our tests show that the approach based on a quadratic formulation is better conditioned and more efficient that the approach based on a linearization. These tests confirm the theoretical results on the conditioning of different barriers, (Güler and Tunçel 1998;Chua and Tunçel 2008).…”

Section: Introductionsupporting

confidence: 78%

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Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

et al. 2008

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We study Semidefinite Programming, SDP , relaxations for Sensor Network Localization, SNL, with anchors and with noisy distance information. The main point of the paper is to view SNL as a (nearest) Euclidean Distance Matrix, EDM, completion problem that does not distinguish between the anchors and the sensors. We show that there are advantages for using the well studied EDM model. In fact, the set of anchors simply corresponds to a given fixed clique for the graph of the EDM problem.We next propose a method of projection when large cliques or dense subgraphs are identified. This projection reduces the size, and improves the stability, of the relaxation. In addition, by viewing the problem as an EDM completion problem, we are able to derive a new approximation scheme for the sensors from the SDP approximation. This yields, on average, better low rank approximations for the low dimensional realizations. This further emphasizes the theme that SNL is in fact just an EDM problem.We solve the SDP relaxations using a primal-dual interior/exterior-point algorithm based on the Gauss-Newton search direction. By not restricting iterations to the interior, we usually get lower rank optimal solutions and thus, better approximations for the SNL problem. We discuss the relative stability and strength of two formulations and the corresponding algorithms that are used. In particular, we show that the quadratic formulation arising from the SDP relaxation is better conditioned than the linearized form that is used in the literature.

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Section: Introductionsupporting

confidence: 78%

“…These tests provide empirical evidence for the theoretical comparison results on different barriers given in Güler and Tunçel (1998), Chua and Tunçel (2008). The results in these references show that the central path is distorted due to the I in the linear formulation constraint.…”

Section: Two Methods For Solving Snlmentioning

confidence: 74%

Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

et al. 2008

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“…These tests provide empirical evidence for the theoretical comparison results on different barriers given in [15,9]. The results in these references show that the central path is distorted due to the I in the linear formulation constraint.…”

Section: Measure 1: Objective Function With Different Anchorsmentioning

confidence: 67%

Sensor network localization, euclidean distance matrix completions, and graph realization

Ding

Krislock

Qian

et al. 2008

Proceedings of the First ACM International Workshop on Mobile Entity Localization and Tracking in GPS-less Environments

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“…These classes of sets have quite different properties in general. We refer the reader to the paper [8] for more details.…”

Section: {0}∪vmentioning

confidence: 99%

Vertices of Spectrahedra arising from the Elliptope, the Theta Body, and Their Relatives

Silva¹,

Tunçel²

2015

SIAM J. Optim.

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Abstract. Utilizing dual descriptions of the normal cone of convex optimization problems in conic form, we characterize the vertices of semidefinite representations arising from Lovász theta body, generalizations of the elliptope and related convex sets. Our results generalize vertex characterizations due to Laurent and Poljak from the 1990's. Our approach also leads us to nice characterizations of strict complementarity and to connections with some of the related literature.

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Invariance and efficiency of convex representations

Cited by 22 publications

References 21 publications

Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

Sensor Network Localization, Euclidean Distance Matrix completions, and graph realization

Sensor network localization, euclidean distance matrix completions, and graph realization

Vertices of Spectrahedra arising from the Elliptope, the Theta Body, and Their Relatives

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