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

Defeng Sun (matsundfnus.edu.sg)
Jie Sun (jsunnus.edu.sg)
Liwei Zhang (lwzhangdlut.edu.cn)

Abstract: 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 $\overline{c}>0$.

Keywords: The augmented Lagrangian method, nonlinear semidefinite programming, rate of convergence, variational analysis

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Nonlinear Optimization

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

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Entry Submitted: 01/25/2006
Entry Accepted: 01/25/2006
Entry Last Modified: 01/25/2006

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