A Newton-CG Augmented Lagrangian Method for Semidefinite Programming∗ Xin-Yuan Zhao
†
Defeng Sun
‡
Kim-Chuan Toh
§
March 12, 2008; Revised, February 03, 2009 Abstract. We consider a Newton-CG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth Newton-CG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts. Keywords: Semidefinite programming, Augmented Lagrangian, Semismoothness, Newton’s method, Iterative solver.
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Introduction
Let S n be the linear space of all n × n symmetric matrices and S+n be the cone of all n × n symmetric positive semidefinite matrices. The notation X 0 means that X is a symmetric ∗
A preliminary version of this paper was presented at ICCOPT II, Hamilton, Canada, August 13, 2007. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 (
[email protected]). ‡ Department of Mathematics and Risk Management Institute, National University of Singapore, 2 Science Drive 2, Singapore 117543 (
[email protected]). This author’s research is supported in part by Academic Research Fund under Grant R-146-000-104-112. § Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543 (
[email protected]); and Singapore-MIT Alliance, 4 Engineering Drive 3, Singapore 117576. 117576. This author’s research is supported in part by Academic Research Grant R-146-000-076-112. †
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positive semidefinite matrix. This paper is devoted to studying an augmented Lagrangian method for solving the following semidefinite programming (SDP) problem n o T ∗ (D) min b y | A y − C 0 , where C ∈ S n , b ∈
