A cyclic projection algorithm via duality

Gaffke, Norbert; Mathar, Rudolf

doi:10.1007/bf02614077

Cited by 115 publications

(108 citation statements)

References 21 publications

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“…Hence, the Method of Alternating Projections (MAP) of Dykstra-Han type [13,22] is a choice. In fact, Glunt et al [17] and Gaffke and Mathar [14] independently studied such MAPs, yet based on different projection formulae onto (−K n + ). A drawback of MAPs is their slow convergence.…”

Section: Introductionmentioning

confidence: 99%

Computing the nearest Euclidean distance matrix with low embedding dimensions

Yuan

2013

Math. Program.

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Euclidean distance embedding appears in many high-profile applications including wireless sensor network localization, where not all pairwise distances among sensors are known or accurate. The classical Multi-Dimensional Scaling (cMDS) generally works well when the partial or contaminated Euclidean Distance Matrix (EDM) is close to the true EDM, but otherwise performs poorly. A natural step preceding cMDS would be to calculate the nearest EDM to the known matrix. A crucial condition on the desired nearest EDM is for it to have a low embedding dimension and this makes the problem nonconvex. There exists a large body of publications that deal with this problem. Some try to solve the problem directly and some are the type of convex relaxations of it. In this paper, we propose a numerical method that aims to solve this problem directly. Our method is strongly motivated by the majorized penalty method of Gao and Sun for low-rank positive semi-definite matrix optimization problems. The basic geometric object in our study is the set of EDMs having a low embedding dimension. We establish a zero duality gap result between the problem and its Lagrangian dual problem, which also motivates the majorization approach adopted. Numerical results show that the method works well for the Euclidean embedding of Network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation.

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Section: Introductionmentioning

confidence: 99%

Computing the nearest Euclidean distance matrix with low embedding dimensions

Yuan

2013

Math. Program.

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“…Note that the feasible region is the intersection of two convex sets and the objective is the orthogonal projection of D onto the feasible region. Therefore, 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64alternating projection methods are natural choices (see [13,16]) and they have been used in molecular conformation [17,18]. Problem (25) can also be reformulated as SDP, see [1] and [44].…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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Abstract:The conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem.For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy.We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP. Response to Reviewers:The author would like to thank the referees and the associate editor for their efforts in their speedy review. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Conditional Quadratic Semidefinite Programming: Examples and MethodsHou-Duo Qi * April 14, 2014; Revised May 3, 2014 AbstractThe conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem. For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy. We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP.

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“…Interestingly, EDM optimization has long been studied in numerical linear algebra and numerical optimization and further applied to molecular conformation, see [17], [11], [12], [13], [24], [25]. What makes EDM optimization more interesting is that such problems can be reformuated as SDP as done in [2], [4], [33].…”

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confidence: 99%

Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

Ding

2016

Comput Optim Appl

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One of the challenging problems in collaborative position localization arises when the distance measurements contain Non-Line-Of-Sight (NLOS) biases. Convex optimization has played a major role in modelling such problems and numerical algorithm developments. One of the successful examples is the Semi-Definite Programming (SDP), which translates Euclidean distances into the constraints of positive semidefinite matrices, leading to a large number of constraints in the case of NLOS biases. In this paper, we propose a new convex optimization model that is built upon the concept of Euclidean Distance Matrix (EDM). The resulting EDM optimization has an advantage that its Lagrangian dual problem is well structured and hence is conducive to algorithm developments. We apply a recently proposed 3-block alternating direction method of multipliers to the dual problem and tested the algorithm on some real as well as simulated data of large scale. In particular, the EDM model significantly outperforms the SDP model and several others. Index TermsEuclidean distance matrix, Collaborative localization, Non-Line of Sight (NLOS), Augmented Lagrangian, Alternating direction method of multipliers (ADMM).

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A cyclic projection algorithm via duality

Cited by 115 publications

References 21 publications

Computing the nearest Euclidean distance matrix with low embedding dimensions

Computing the nearest Euclidean distance matrix with low embedding dimensions

Conditional Quadratic Semidefinite Programming: Examples and Methods

Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

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