Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems

Pang, Jong-Shi; Sun, Defeng; Sun, Jie

doi:10.1287/moor.28.1.39.14258

Cited by 114 publications

(78 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The relation (21) suggests the following approach: first solve the unconstrained convex optimization problem (15) for y * and then obtain the unique solution to the original problem (13) by (21). This is exactly the well-known Lagrangian dual approach outlined by Rockafellar [26].…”

Section: Discussionmentioning

confidence: 99%

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

Sun

2009

Comput Optim Appl

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

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

Sun

2009

Comput Optim Appl

Self Cite

View full text Add to dashboard Cite

show abstract

“…Since (y, X) satisfies (12), we can assume that there exists an orthogonal matrix P ∈ n×n such that X and (A * y − C) have the spectral decompositions as in (14). From (14), (22), and (31), we have…”

Section: Thusmentioning

confidence: 99%

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

Zhao¹,

Sun²,

Toh³

2010

SIAM J. Optim.

Self Cite

365

387

View full text Add to dashboard Cite

Abstract. We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth Newton-CG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts.

show abstract

“…It follows from [23,Lemma 11] that V ∈ ∂Φ(X) and also V ∈ ∂Φ(X). This type of matrices have been previously used in implementing Newton's method in [26,4] for the nearest correlation matrix problem.…”

Section: Generalized Jacobian Ofmentioning

confidence: 99%

A Convex Matrix Optimization for the Additive Constant Problem in Multidimensional Scaling with Application to Locally Linear Embedding

Qi¹

2016

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract. The additive constant problem has a long history in multi-dimensional scaling and it has recently been used to resolve the issue of indefiniteness of the geodesic distance matrix in ISOMAP. But it would lead to a large positive constant being added to all eigenvalues of the centered geodesic distance matrix, often causing significant distortion of the original distances. In this paper, we reformulate the problem as a convex optimization of almost negative semidefinite matrix so as to achieve minimal variation of the original distances. We then develop a Newton-CG method and further prove its quadratic convergence. Finally, we include a novel application to the famous LLE (Locally Linear Embedding in nonlinear dimensionality reduction), addressing the issue when the input of LLE has missing values. We justify the use of the developed method to tackle this issue by establishing that the local Gram matrix used in LLE can be obtained through a local Euclidean distance matrix. The effectiveness of our method is demonstrated by numerical experiments

show abstract

Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems

Cited by 114 publications

References 44 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

A Newton-CG Augmented Lagrangian Method for Semidefinite Programming

A Convex Matrix Optimization for the Additive Constant Problem in Multidimensional Scaling with Application to Locally Linear Embedding

Contact Info

Product

Resources

About