We consider the sequence acceleration problem for the alternating direction method of multipliers (ADMM) applied to a class of equality-constrained problems with strongly convex quadratic objectives, which frequently arise as the Newton subproblem of interior-point methods. Within this context, the ADMM update equations are linear, the iterates are confined within a Krylov subspace, and the general minimum residual (GMRES) algorithm is optimal in its ability to accelerate convergence. The basic ADMM method solves a $\kappa$-conditioned problem in $O(\sqrt{\kappa})$ iterations. We give theoretical justification and numerical evidence that the GMRES-accelerated variant consistently solves the same problem in $O(\kappa^{1/4})$ iterations for an order-of-magnitude reduction in iterations, despite a worst-case bound of $O(\sqrt{\kappa})$ iterations. The method is shown to be competitive against standard preconditioned Krylov subspace methods for saddle-point problems. The method is embedded within SeDuMi, a popular open-source solver for conic optimization written in MATLAB, and used to solve many large-scale semidefinite programs with error that decreases like $O(1/k^{2})$, instead of $O(1/k)$, where $k$ is the iteration index.


  1. ADMM
  2. alternating direction
  3. method of multipliers
  4. augmented Lagrangian
  5. sequence acceleration
  6. GMRES
  7. Krylov subspace

MSC codes

  1. 49M20
  2. 90C06
  3. 65B99

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Information & Authors


Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 3025 - 3056
ISSN (online): 1095-7189


Submitted: 4 February 2016
Accepted: 28 August 2018
Published online: 25 October 2018


  1. ADMM
  2. alternating direction
  3. method of multipliers
  4. augmented Lagrangian
  5. sequence acceleration
  6. GMRES
  7. Krylov subspace

MSC codes

  1. 49M20
  2. 90C06
  3. 65B99



Funding Information

Defense Advanced Research Projects Agency https://doi.org/10.13039/100000185
Air Force Office of Scientific Research https://doi.org/10.13039/100000181
Office of Naval Research https://doi.org/10.13039/100000006
Skolkovo Institute of Science and Technology https://doi.org/10.13039/501100007455

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