Convergence analysis of some algorithms for solving nonsmooth equations
Mathematics of Operations Research
Complementarity and nondegeneracy in semidefinite programming
Mathematical Programming: Series A and B
SIAM Journal on Optimization
Semismooth Matrix-Valued Functions
Mathematics of Operations Research
Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems
Mathematics of Operations Research
A Dual Approach to Semidefinite Least-Squares Problems
SIAM Journal on Matrix Analysis and Applications
Least-Squares Covariance Matrix Adjustment
SIAM Journal on Matrix Analysis and Applications
A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix
SIAM Journal on Matrix Analysis and Applications
Mathematics of Operations Research
A Dual Optimization Approach to Inverse Quadratic Eigenvalue Problems with Partial Eigenstructure
SIAM Journal on Scientific Computing
The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming
Mathematical Programming: Series A and B
Calibrating Least Squares Semidefinite Programming with Equality and Inequality Constraints
SIAM Journal on Matrix Analysis and Applications
Computing a Nearest Correlation Matrix with Factor Structure
SIAM Journal on Matrix Analysis and Applications
A projected semismooth Newton method for problems of calibrating least squares covariance matrix
Operations Research Letters
An inexact spectral bundle method for convex quadratic semidefinite programming
Computational Optimization and Applications
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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.