Approximation by matrices positive semidefinite on a subspace

Hayden, T. L.; Wells, J. H.

doi:10.1016/0024-3795(88)90202-9

Cited by 50 publications

(39 citation statements)

References 10 publications

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“…Problem (4) was studied by Hayden and Wells [21]. Like the positive semidefiniteness case (2), it also has a closed form solution.…”

Section: Generalized Matrix Approximationmentioning

confidence: 99%

“…Hence, Y * is a feasible solution of (21). In the meantime, given that (22) is a relaxation of (21) we have…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

“…In literature, model (22) is also often referred to as the cMDS, which intends to solve model (21). Thm.…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

“…Unlike cMDS model (21), this problem has no closed-form solution anymore. Hence, iterative algorithms have to be used.…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

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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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“…Problem (4) was studied by Hayden and Wells [21]. Like the positive semidefiniteness case (2), it also has a closed form solution.…”

Section: Generalized Matrix Approximationmentioning

confidence: 99%

“…Hence, Y * is a feasible solution of (21). In the meantime, given that (22) is a relaxation of (21) we have…”

Section: Nearest Edm Problemsmentioning

confidence: 99%

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Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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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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Dykstra's Algorithm for a Constrained Least-squares Matrix Problem

Escalante

Raydan

1996

Numer. Linear Algebra Appl.

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We apply Dykstra's alternating projection algorithm to the constrained least‐squares matrix problem that arises naturally in statistics and mathematical economics. In particular, we are concerned with the problem of finding the closest symmetric positive definite bounded and patterned matrix, in the Frobenius norm, to a given matrix. In this work, we state the problem as the minimization of a convex function over the intersection of a finite collection of closed and convex sets in the vector space of square matrices. We present iterative schemes that exploit the geometry of the problem, and for which we establish convergence to the unique solution. Finally, we present preliminary numberical results to illustrate the performance of the proposed iterative methods.

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Approximation by matrices positive semidefinite on a subspace

Cited by 50 publications

References 10 publications

Conditional Quadratic Semidefinite Programming: Examples and Methods

Conditional Quadratic Semidefinite Programming: Examples and Methods

Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

Dykstra's Algorithm for a Constrained Least-squares Matrix Problem

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