Parameter Insensitivity in ADMMPreconditioned Solution of SaddlePoint Problems
Abstract
We consider the solution of linear saddlepoint problems, using the alternating direction methodofmultipliers (ADMM) as a preconditioner for the generalized minimum residual method (GMRES). We show, using theoretical bounds and empirical results, that ADMM is made remarkably insensitive to the parameter choice with Krylov subspace acceleration. We prove that ADMMGMRES can consistently converge, irrespective of the exact parameter choice, to an $\epsilon$accurate solution of a $\kappa$conditioned problem in $O(\kappa^{2/3}\log\epsilon^{1})$ iterations. The accelerated method is applied to randomly generated problems, as well as the Newton direction computation for the interiorpoint solution of semidefinite programs in the SDPLIB test suite. The empirical results confirm this parameter insensitivity, and suggest a slightly improved iteration bound of $O(\sqrt{\kappa}\log\epsilon^{1})$.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 arXiv:
 arXiv:1602.02135
 Bibcode:
 2016arXiv160202135Z
 Keywords:

 Mathematics  Optimization and Control
 EPrint:
 20 pages, 8 figures