An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem

Qi, Houduo; Sun, Defeng

doi:10.1093/imanum/drp031

Cited by 46 publications

(51 citation statements)

References 43 publications

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“…3.2] 16 Proposition 4.4 For any given h ∈ IR n . Let Q(A * (h))Q T have the partition in (35). For every matrix V ∈ ∂F (y), there exists V ∈ ∂Π S |β|+r + (0) such that…”

Section: Characterization Of ∂F (Y)mentioning

confidence: 99%

“…We have the following characterization of constraint nondegeneracy. 17 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 64 65 Proposition 4.5 Let h ∈ IR m be given and Q T (A * (h))Q have the representation of (35). Let A ∈ K n + (V) be decomposed as in (37) and the resulting Z 1 have the spectral decomposition (38).…”

Section: Nonsingularity Of ∂F (Y)mentioning

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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“…3.2] 16 Proposition 4.4 For any given h ∈ IR n . Let Q(A * (h))Q T have the partition in (35). For every matrix V ∈ ∂F (y), there exists V ∈ ∂Π S |β|+r + (0) such that…”

Section: Characterization Of ∂F (Y)mentioning

confidence: 99%

Section: Nonsingularity Of ∂F (Y)mentioning

confidence: 99%

Conditional Quadratic Semidefinite Programming: Examples and Methods

2014

J. Oper. Res. Soc. China

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“…This is in contrast to the nearest correlation matrix problem in the Frobenius norm, where the so-called H-weighted problem min{ W • (M 0 − X) F : X is a correlation matrix } is much more challenging to solve than the unweighted problem with w ij = 1 [18], [31]. For example, the NAG code nag_nearest_correlation_h_weight (g02aj) solves the H-weighted nearest correlation matrix problem, but the documentation says that if the weights vary by several orders of magnitude then the underlying algorithm may fail to converge [28].…”

Section: Introducing Weightsmentioning

confidence: 99%

Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block

Higham

Strabić²,

Šego³

2016

SIAM Rev.

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Abstract. Indefinite approximations of positive semidefinite matrices arise in various data analysis applications involving covariance matrices and correlation matrices. We propose a method for restoring positive semidefiniteness of an indefinite matrix M 0 that constructs a convex linear combination S(α) = αM 1 + (1 − α)M 0 of M 0 and a positive semidefinite target matrix M 1 . In statistics, this construction for improving an estimate M 0 by combining it with new information in M 1 is known as shrinking. We make no statistical assumptions about M 0 and define the optimal shrinking parameter as α * = min{α ∈ [0, 1] : S(α) is positive semidefinite}. We describe three algorithms for computing α * . One algorithm is based on the bisection method, with the use of Cholesky factorization to test definiteness; a second employs Newton's method; and a third finds the smallest eigenvalue of a symmetric definite generalized eigenvalue problem. We show that weights that reflect confidence in the individual entries of M 0 can be used to construct a natural choice of the target matrix M 1 . We treat in detail a problem variant in which a positive semidefinite leading principal submatrix of M 0 remains fixed, showing how the fixed block can be exploited to reduce the cost of the bisection and generalized eigenvalue methods. Numerical experiments show that when applied to indefinite approximations of correlation matrices shrinking can be at least an order of magnitude faster than computing the nearest correlation matrix.

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“…When M n + reduces to S n + , problem (1) becomes the semidefinite least squares problem (SDLS) which has many applications in finance, insurance and reinsurence. There have been extensive studies on the SDLS recently, see Higham [6], Malick [10], Boyd and Xiao [4], Qi and Sun [16,18], Toh [23] and Borsdorf and Higham [2], etc. An important approach that emerged from these study is the Lagrangian dual approach, originated by Rockafellar and exemplified for SDLS in the above papers.…”

Section: Introductionmentioning

confidence: 99%

A regularized strong duality for nonsymmetric semidefinite least squares problem

2010

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The nonsymmetric semidefinite least squares problem (NSDLS) is to find a nonsymmetric semidefinite matrix which is closest to a given matrix in Frobenius norm. It is an extension of the semidefinite least squares problem (SDLS) and has important application in the area of robotics and automation. In this note, by developing the minimal representation of the underlying cone with the linear constraints, we obtain a regularized strong duality with low-dimensional projection for NSDLS. Further, we study the generalized differential properties and nonsingularity of the first order optimality system about the dual problem. These theoretical results demonstrate that we can solve NSDLS as good as the current Lagrangian dual approaches to SDLS.

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An augmented Lagrangian dual approach for the H-weighted nearest correlation matrix problem

Cited by 46 publications

References 43 publications

Conditional Quadratic Semidefinite Programming: Examples and Methods

Conditional Quadratic Semidefinite Programming: Examples and Methods

Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block

A regularized strong duality for nonsymmetric semidefinite least squares problem

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