Local Duality of Nonlinear Semidefinite Programming
Mathematics of Operations Research
Dualize it: software for automatic primal and dual conversions of conic programs
Optimization Methods & Software
A Newton-CG Augmented Lagrangian Method for Semidefinite Programming
SIAM Journal on Optimization
Block relaxation and majorization methods for the nearest correlation matrix with factor structure
Computational Optimization and Applications
Solving Large-Scale Least Squares Semidefinite Programming by Alternating Direction Methods
SIAM Journal on Matrix Analysis and Applications
A Sequential Semismooth Newton Method for the Nearest Low-rank Correlation Matrix Problem
SIAM Journal on Optimization
An inexact spectral bundle method for convex quadratic semidefinite programming
Computational Optimization and Applications
Approximation of rank function and its application to the nearest low-rank correlation matrix
Journal of Global Optimization
Hi-index | 0.00 |
We propose primal–dual path-following Mehrotra-type predictor–corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: $$\min_{X} \{\frac{1}{2} X\bullet \mathcal{Q}(X) + C\bullet X : \mathcal{A} (X) = b, X\succeq 0\}$$, where $$\mathcal{Q}$$ is a self-adjoint positive semidefinite linear operator on $$\mathcal{S}^n$$, b∈R m , and $$\mathcal{A}$$ is a linear map from $$\mathcal{S}^n$$ to R m . At each interior-point iteration, the search direction is computed from a dense symmetric indefinite linear system (called the augmented equation) of dimension m + n(n + 1)/2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust.