The Rate of Convergence of the Augmented Lagrangian Method for Nonlinear Semidefinite Programming∗ Defeng Sun†, Jie Sun‡, and
Liwei Zhang§
March 31, 2006
Abstract We analyze the local convergence rate 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 > 0. Key words: The augmented Lagrangian method, nonlinear semidefinite programming, rate of convergence, variational analysis.
1
Introduction
The nonconvex semidefinite programming problem has wide applications in system control, structural design, and other fields. It has recently become a focal point in optimization research. For example, in the recent release of the library COMPle ib [22], a total of 168 test examples for nonlinear semidefinite programs, control system design, and related problems are collected. Among very few algorithms for this problem, the augmented Lagrangian method appears to perform well [24]. It naturally calls for a suitable theoretical explanation for this phenomenon. In its general setting, the augmented Lagrangian method can be used to solve the following optimization problem ∗
The research of Defeng Sun is partly supported by Grant R146-000-061-112 from the National University of Singapore. The research of Jie Sun and Liwei Zhang is partly supported by Singapore-MIT Alliance and by Grants RP314000-042/057-112 of the National University of Singapore. The research of Liwei Zhang is also supported by the National Natural Science Foundation of China under project grant No. 10471015 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. † Department of Mathematics, National University of Singapore, Republic of Singapore (email:
[email protected]). ‡ Department of Decision Sciences, National University of Singapore, Republic of Singapore (email:
[email protected]). § Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China (email:
[email protected]).
1
(OP)
min f (x)
s.t.
h(x) = 0, g(x) ∈ K ,
where f : X 7→
