ACM Transactions on Mathematical Software (TOMS)
Distributed large scale network utility maximization
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2
A Newton-CG Augmented Lagrangian Method for Semidefinite Programming
SIAM Journal on Optimization
ACM Transactions on Mathematical Software (TOMS)
A trust region method for solving semidefinite programs
Computational Optimization and Applications
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The search directions in an interior-point method for large scale semidefinite programming (SDP) can be computed by applying a Krylov iterative method to either the Schur complement equation (SCE) or the augmented equation. Both methods suffer from slow convergence as interior-point iterates approach optimality. Numerical experiments have shown that a diagonally preconditioned conjugate residual method on the SCE typically takes a huge number of steps to converge. However, it is difficult to incorporate cheap and effective preconditioners into the SCE. This paper proposes to apply the preconditioned symmetric quasi-minimal residual (PSQMR) method to a reduced augmented equation that is derived from the augmented equation by utilizing the eigenvalue structure of the interior-point iterates. Numerical experiments on SDP problems arising from maximum clique and selected SDPLIB (SDP Library) problems show that moderately accurate solutions can be obtained with a modest number of PSQMR steps using the proposed preconditioned reduced augmented equation. An SDP problem with 127600 constraints is solved in about 6.5 hours to an accuracy of 10-6 in relative duality gap.