A nonsmooth version of Newton's method
Mathematical Programming: Series A and B
A globally convergent Newton method for convex SC1minimization problems
Journal of Optimization Theory and Applications
Matrix computations (3rd ed.)
Complementarity and nondegeneracy in semidefinite programming
Mathematical Programming: Series A and B
Solving Some Large Scale Semidefinite Programs via the Conjugate Residual Method
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
Solving Large Scale Semidefinite Programs via an Iterative Solver on the Augmented Systems
SIAM Journal on Optimization
Solving Lift-and-Project Relaxations of Binary Integer Programs
SIAM Journal on Optimization
A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix
SIAM Journal on Matrix Analysis and Applications
On the solution of large-scale SDP problems by the modified barrier method using iterative solvers
Mathematical Programming: Series A and B
An inexact primal–dual path following algorithm for convex quadratic SDP
Mathematical Programming: Series A and B
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
Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming
SIAM Journal on Optimization
An Augmented Primal-Dual Method for Linear Conic Programs
SIAM Journal on Optimization
Regularization Methods for Semidefinite Programming
SIAM Journal on Optimization
Copositive and semidefinite relaxations of the quadratic assignment problem
Discrete Optimization
Minimization of SC1 functions and the Maratos effect
Operations Research Letters
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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 SDP problems 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 in previous attempts.