Abstract
In this paper, we propose a partially parallel prediction-correction splitting method for solving block-separable linearly constrained convex optimization problems with three blocks. Unlike the extended alternating direction method of multipliers, the last two subproblems in the prediction step are solved parallelly, and a correction step is employed in the method to correct the dual variable and two blocks of the primal variables. The step size adapted in the correction step allows for major contribution from the latest solution point to the iteration point. Some numerical results are reported to show the effectiveness of the presented method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, Boston (1982)
Bertsekas, D.P., Gafni, E.M.: Projection method for variational inequalities with applications to the traffic assignment problem. Math. Progr. Stud. 17, 139–159 (1982)
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation, Numerical Methods. Prentice Hall, Englewood Cliffs (1989)
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2010)
Han, D., Kong, W., Zhang, W.: A partial splitting augmented lagrangian method for low patch-rank image decomposition. J. Math. Imaging Vis. 51, 145–160 (2015)
Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Comput. Math. Appl. 2, 17–40 (1976)
Glowinski, R., Marrocco, A.: Sur l’ approximation par éléments nis d’ordre un, et la résolution par pénalisation-dualité d’une classe de problémes de Dirichlet nonlinéaires. J Equine Vet Sci 2, 41–76 (1975)
Han, D., Yuan, X.: A note on the alternating direction method of multipliers. J. Optim. Theory Appl. 155, 227–238 (2012)
Han, D., Yuan, X., Zhang, W., Cai, X.: An ADM-based splitting methods for separable convex programming. Comput. Optim. Appl. 54, 343–369 (2013)
He, B., Tao, M., Xu, M., Yuan, X.: Alternating direction-based contraction method for generally separable linearly constrained convex programming problems. Optimization 62, 573–596 (2013)
He, B., Tao, M., Yuan, X.: Alternating direction method with gaussian back substitution for separable convex programming. SIAM J. Optim. 22, 313–340 (2012)
Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Lian-Sheng Zhang in celebration of his 80th birthday.
Rights and permissions
About this article
Cite this article
Bai, FS., Xu, L. A Partially Parallel Prediction-Correction Splitting Method for Convex Optimization Problems with Separable Structure. J. Oper. Res. Soc. China 5, 529–544 (2017). https://doi.org/10.1007/s40305-017-0163-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40305-017-0163-5
Keywords
- Block-separable convex optimization
- Extended alternating direction method of multipliers
- Prediction–correction splitting method