Further Study on the Convergence Rate of Alternating Direction Method of Multipliers with Logarithmic-quadratic Proximal Regularization
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In the literature, the combination of the alternating direction method of multipliers with the logarithmic-quadratic proximal regularization has been proved to be convergent, and its worst-case convergence rate in the ergodic sense has been established. In this paper, we focus on a convex minimization model and consider an inexact version of the combination of the alternating direction method of multipliers with the logarithmic-quadratic proximal regularization. Our primary purpose is to further study its convergence rate and to establish its worst-case convergence rates measured by the iteration complexity in both the ergodic and non-ergodic senses. In particular, existing convergence rate results for this combination are subsumed by the new results.
KeywordsConvex programming Alternating direction method of multipliers Logarithmic-quadratic proximal Convergence rate Iteration complexity
Mathematics Subject Classification90C25 90C33 65K05
The first author was supported in part by the Natural Science Foundation of Jiangsu Province Grant BK20130550 and the NSFC Grant 11401300. The second author was supported by the Program for New Century Excellent Talents in University Grant NCET-12-0111, Qing Lan Project and the Natural Science Foundation of Jiangsu Province Grant BK2012662. The third author was partially supported by the Grant from Hong Kong Baptist University: FRG2/13-14/061 and the General Research Fund from Hong Kong Research Grants Council: 203613.
- 1.Glowinski, R., Marroco, A.: Sur l’approximation, paréléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires. ESAIM: Math. Model. Numer. Anal. – Modélisation Mathématique et Analyse Numérique 9(R2), 41–76 (1975)Google Scholar
- 4.Eckstein, J., Yao W.: Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results. RUTCOR Research Report RRR 32-2012 (2012)Google Scholar
- 5.Glowinski, R.: On alternating direction methods of multipliers: a historical perspective. In: Fitzgibbon, W., Kuznetsov, Yu. A., Neittaanmaki, P., Pironneau, O. (eds.), Modeling, Simulation and Optimization for Science and Technology. Computational Methods in Applied Sciences, vol. 34, pp. 59–82. Springer, Dordrecht (2014)Google Scholar
- 11.He, B.S., Liao, L-Z, Yuan, X.M.: A LQP based interior prediction-correction method for nonlinear complementarity problems. J. Comput. Math. 24(1), 33–44 (2006)Google Scholar
- 15.Li, M., Liao, L.-Z, Yuan, X.M.: Inexact alternating direction method of multipliers with logarithmic-quadratic proximal regularization. J. Optim. Theory Appl. 159, 412–436 (2013)Google Scholar
- 17.He, B.S., Yuan, X.M.: On nonergodic convergence rate of Douglas-Rachford alternating direction method of multipliers. Numer. Math. under minor revision.Google Scholar
- 20.Deng, W., Lai, M.J., Peng, Z.M., Yin, W.T.: Parallel multi-block ADMM with \(o(1/k)\) convergence. manuscript (2013)Google Scholar
- 22.Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research. Springer, New York (2003)Google Scholar
- 24.Li, M., Yuan, X.M.: A strictly contractive Peaceman-Rachford splitting method with logarithmic-quadratic proximal regularization for convex programming. to appear in Math. Oper. Res. (2014).Google Scholar