Abstract
We propose an iterative scheme for solving variational inequalities with monotone operators over affine sets in an infinite dimensional Hilbert space setting. We show that several primal-dual algorithms in the literature as well as the classical ADMM algorithm for convex optimization problems, together with some of its variants, are encompassed by the proposed numerical scheme. Furthermore, we carry out a convergence analysis of the generated iterates and provide convergence rates by using suitable dynamical step sizes together with variable metric techniques.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Banert, S., Boţ, R.I., Csetnek, E.R.: Fixing and extending some recent results on the ADMM algorithm (2017). Available via arXiv. https://arxiv.org/abs/1612.05057
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics, Springer, New York (2011)
Boţ, R.I.: Conjugate Duality in Convex Optimization. Lecture Notes in Economics and Mathematical Systems 637, Springer-Verlag Berlin Heidelberg (2010)
Boţ, R.I., Csetnek, E.R.: ADMM for monotone operators: convergence analysis and rates. Adv. Comput. Math. 45(1), 327–359 (2019). Available via arXiv. https://arxiv.org/abs/1705.01913
Boţ, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximal monotone operators. SIAM J. Optim. 23, 2011–2036 (2013)
Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning 3, 1–12 (2010)
Chambolle, P.L., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40, 120–145 (2011)
Combettes, P.L., Vũ, B.C.: Variable metric quasi-Feyér monotonicity. Nonlinear Anal. 78, 17–31 (2013)
Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158, 460–479 (2013)
Fazel, M., Pong, T.K., Sun, D., Tseng, P.: Hankel matrix rank minimization with applications in system identification and realization. SIAM J. Matrix Anal. 34, 946–977 (2013)
Fortin, M., Glowinski, R.: On decomposition-coordination methods using an augmented Lagrangian. In: Fortin, M. and Glowinski, R. (eds.), Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, North-Holland, Amsterdam (1983)
Gabay, D.: Applications of the method of multipliers to variational inequalities. In: Fortin, M. and Glowinski, R. (eds.), Augmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems, North-Holland, Amsterdam (1983)
Riesz, F., Nagy, B.Sz.: Leo̧ns d’Analyse Fonctionelle. Fifth ed., Gauthier-Villars, Paris (1968)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pacific J. Math. 33, 209–216 (1970)
Shefi, R., Teboulle, M.: Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization. SIAM J. Optim. 24, 269–297 (2014)
Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667–681 (2013)
Acknowledgements
The first author has been partially supported by FWF (Austrian Science Fund), project I 2419-N32. The second author has been supported by FWF (Austrian Science Fund), project P 29809-N32. The third author has been partially supported by FWF (Austrian Science Fund), project I 2419-N32, and by the Doctoral Programme Vienna Graduate School on Computational Optimization (VGSCO), project W1260-N35.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Boţ, R.I., Csetnek, E.R., Meier, D. (2019). Variable Metric ADMM for Solving Variational Inequalities with Monotone Operators over Affine Sets. In: Bauschke, H., Burachik, R., Luke, D. (eds) Splitting Algorithms, Modern Operator Theory, and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-25939-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-25939-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-25938-9
Online ISBN: 978-3-030-25939-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)