The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

Sun, Defeng; Sun, Jie; Zhang, Liwei

doi:10.1007/s10107-007-0105-9

Cited by 134 publications

(92 citation statements)

References 40 publications

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“…By using the facts that C is positive definite and A : S n → R (n+κ) is onto, we know that the generalized Slater condition (20) for problem (32) holds. Thus, Algorithm 3.1 will generate a bounded sequence {y k } when it is applied to problem (33).…”

Section: The Band Correlation Stress Testingmentioning

confidence: 99%

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: The Band Correlation Stress Testingmentioning

confidence: 99%

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

Sun

2009

Comput Optim Appl

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“…Interestingly, Theorem 3.4 implies that the sequence {(y k+1 , Z k+1 )} converges to (ȳ, Z ) at a linear rate that is inversely proportional to c k for all c k sufficiently large. This fast convergence has a recent new interpretation in the context of NSDP: locally, the augmented Lagrangian method can be treated as an approximate semismooth Newton method (see Sun et al, 2008) for the equation…”

Section: Outline Of the Augmented Lagrangian Methodsmentioning

confidence: 99%

“…In fact, it was proven in Sun et al (2008) (in the NSDP setting) that, for any c large enough, ∇ν c is semismooth at (ȳ, Z ) and one has the following estimate:…”

Section: Outline Of the Augmented Lagrangian Methodsmentioning

confidence: 99%

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

Sun

2010

IMA Journal of Numerical Analysis

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Higham (2002, IMA J. Numer. Anal., 22, 329-343) considered two types of nearest correlation matrix problems, namely the W -weighted case and the H -weighted case. While the W -weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H -weighted case remains numerically challenging. The difficulty of extending those methods from the W -weighted case to the H -weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H -weight, unlike the case under the W -weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H -weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.

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“…It bases on generalized augmented Lagrangians designed for the semidefinite constraint and solves a sequence of unconstrained minimization problems driven by a penalty parameter. There are other approaches for dealing with general NLSDP, for instance, sequential semidefinite programming [46,48,36,56,51,52,131,139,43], bundle methods [104,105], partially augmented Lagrangian approach [10,45,106], interior point trust region [89,90,91], predictor-corrector interior point [64], augmented Lagrangian [106,132], successive linearization [68] and primal-dual interior point methods [142,143,69] among others. There is not a definitive answer to the question of which is the most convenient approach for solving NLSDP in general, which explains the intense research activity going on in this area.…”

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

Linear and Nonlinear Semidefinite Programming

Bofill

Gómez

2014

Pesqui. Oper.

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ABSTRACT. This paper provides a short introduction to optimization problems with semidefinite constraints. Basic duality and optimality conditions are presented. For linear semidefinite programming some advances by dealing with degeneracy and the semidefinite facial reduction are discussed. Two relatively recent areas of application are presented. Finally a short overview of relevant literature on algorithmic approaches for efficiently solving linear and nonlinear semidefinite programming is provided.

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The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

Cited by 134 publications

References 40 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

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

Linear and Nonlinear Semidefinite Programming

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