The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

Indexed by:Journal Article

Date of Publication:2008-08-01

Journal:MATHEMATICAL PROGRAMMING

Included Journals:Scopus、EI、SCIE

Volume:114

Issue:2

Page Number:349-391

ISSN:0025-5610

Key Words:nonlinear semidefinite programming; rate of convergence; the augmented Lagrangian method; variational analysis

Abstract:We analyze the rate of local convergence of the augmented Lagrangian method in 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 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 of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold (c) over bar > 0.