Local Duality of Nonlinear Semidefinite Programming
Mathematics of Operations Research
Calibrating Least Squares Semidefinite Programming with Equality and Inequality Constraints
SIAM Journal on Matrix Analysis and Applications
A Newton-CG Augmented Lagrangian Method for Semidefinite Programming
SIAM Journal on Optimization
Solving Log-Determinant Optimization Problems by a Newton-CG Primal Proximal Point Algorithm
SIAM Journal on Optimization
A projected semismooth Newton method for problems of calibrating least squares covariance matrix
Operations Research Letters
Nonsingularity Conditions for the Fischer-Burmeister System of Nonlinear SDPs
SIAM Journal on Optimization
On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming
Journal of Global Optimization
Nonsingularity of FB system and constraint nondegeneracy in semidefinite programming
Numerical Algorithms
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It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, and the nonsingularity of the corresponding Clarke's generalized Jacobian, at a KKT point, are all equivalent. Moreover, we prove the equivalence between each of these conditions and the nonsingularity of Clarke's generalized Jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing Newton methods. In particular, we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies, but without the strict complementarity.