Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems

Chen, Xin; Qi, Houduo; Tseng, Paul

doi:10.1137/s1052623400380584

Cited by 77 publications

(91 citation statements)

References 33 publications

(101 reference statements)

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“…As a matter of fact, the classical Newton's method is invalid in this situation as the Hessian of θ(·) at some points may not exist at all. Fortunately, the recent study conducted by Qi and Sun [22] for the nearest correlation matrix problem (8) indicates that one may still expect a quadratically converging Newton's method by using the fact S p + is strongly semismooth everywhere in S p , a key property proven by Sun and Sun [31] and extended by Chen, Qi, and Tseng [8] to some more general matrix valued functions.…”

Section: Discussionmentioning

confidence: 99%

“…Given the above preparations, we can extend directly the generalized Newton method developed in [22] from the nearest correlation problem (8) to problem (15) with ∂ 2 θ(·) being replaced by ∂ 2 θ(·).…”

Section: ∂F (Y) = Conv ∂ B F (Y)mentioning

confidence: 99%

“…These examples include: a 7-factor example in Finger [13], a 4-factor example in Turkay, Epperlein, and Christofides [34], a 12-factor example in Rebonato and Jäckel [25], and a 5-factor example in Bhansali and Wise [5]. For all these 8 Theoretically speaking, in order to make Algorithm 4.3 practical, one should consider the case that…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

Sun

2009

Comput Optim Appl

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Correlation stress testing is employed in several financial models for determining the value-at-risk (VaR) of a financial institution's portfolio. The possible lack of mathematical consistence in the target correlation matrix, which must be positive semidefinite, often causes breakdown of these models. The target matrix is obtained by fixing some of the correlations (often contained in blocks of submatrices) in the current correlation matrix while stressing the remaining to a certain level to reflect various stressing scenarios. The combination of fixing and stressing effects often leads to mathematical inconsistence of the target matrix. It is then naturally to find the nearest correlation matrix to the target matrix with the fixed correlations unaltered. However, the number of fixed correlations could be potentially very large, posing a computational challenge to existing methods. In this paper, we propose an unconstrained convex optimization approach by solving one or a sequence of continuously differentiable (but not twice continuously differentiable) convex optimization problems, depending on different stress patterns. This research fully takes advantage of the recently developed theory of strongly semismooth matrix valued functions, which makes fast convergent numerical methods applicable to the underlying unconstrained optimization problem. Promising numerical results on practical data (RiskMetrics database) and randomly generated problems of larger sizes are reported.

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Section: Discussionmentioning

confidence: 99%

Section: ∂F (Y) = Conv ∂ B F (Y)mentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

Sun

2009

Comput Optim Appl

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“…Moreover, it is strongly semismooth due to Sun and Sun [43] (see also Chen et al [6]). The strong semismoothness of Π S n + (·) is a fundamental property behind semismooth Newton-CG methods for matrix optimization problems, see Qi and Sun [34] and Zhao et al [49].…”

Section: Matrix Approximation and Completion On A Subspacementioning

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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“…Strong semismoothness plays a fundamental role in the analysis of the quadratic convergence of Newton's method for solving systems of nonsmooth equations [13,14]. Newton-type methods for solving the semidefinite programming and the semidefinite complementarity problem based on a smoothed form of Φ sdc min are discussed in [4,5,12,17]. is its continuous differentiability [18].…”

Section: Introductionmentioning

confidence: 99%

Strong Semismoothness of the Fischer-Burmeister SDC and SOC Complementarity Functions

Sun

2005

Math. Program.

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Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems

Cited by 77 publications

References 33 publications

Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

Conditional Quadratic Semidefinite Programming: Examples and Methods

Strong Semismoothness of the Fischer-Burmeister SDC and SOC Complementarity Functions

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