# The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming

We analyze the rate of local convergence of the augmented Lagrangian method for nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and certain variational analysis on the projection operator in the symmetric-matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate is proportional to $1/c$, where $c$ is the penalty parameter that exceeds a threshold $\overline{c}>0$.

## Citation

Technical Report, Department of Mathematics, National University of Singapore, January 2006.