A Newton-CG Augmented Lagrangian Method for Semidefinite Programming
Name
Zhao-2010-A NEWTON-CG AUGMENTE.pdf
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Author(s)
Toh, Kim Chuan
Zhao, Xin-Yuan
Sun, Defeng
Alternative Title
A NEWTON-CG AUGMENTED LAGRANGIAN METHOD FOR SEMIDEFINITE PROGRAMMING
Date Issued
January 2010
Journal
SIAM Journal on Optimization
Publisher
Society for Industrial and Applied Mathematics
Citation
Zhao, Xin-Yuan, Defeng Sun, and Kim-Chuan Toh. "A Newton-CG Augmented Lagrangian Method for Semidefinite Programming." SIAM J. Optim. Volume 20, Issue 4, pp. 1737-1765 (2010) ©2010 Society for Industrial and Applied Mathematics.
Version
Final published version
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 SDP problems 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 in previous attempts.
MIT Department
Singapore-MIT Alliance in Research and Technology (SMART)
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DOI of Published Version
https://doi.org/10.1137/080718206
