An Inertial Forward-backward Splitting Method for Monotone Inclusions
Main Article Content
Abstract
In this work, we propose a splitting method for solving monotone inclusions in Hilbert spaces. Our method is a modification of the forward-backward algorithm by using the inertial effect. The weak convergence of the proposed algorithm is established under standard conditions.
Keywords:
Monotone inclusion, Splitting method, Inertial effect, Forward-backward algorithm. Mathematics Subject Classifications (2020): 47H05, 49M29, 49M27, 90C25..
References
K. Aoyama, Y. Kimura, W. Takahashi, Maximal Monotone Operators and Maximal Monotone Functions for Equilibrium Problems, J. Convex Anal., Vol. 15, 2008, pp. 395-409.
M. N. Bui, P. L. Combettes, Multivariate Monotone Inclusions in Saddle Form. Math. Oper. Res., Vol. 47, No. 2, 2022, pp. 1082-1109.
P. L. Combettes, J. C. Pesquet. Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators, Set-Valued Var. Anal., Vol. 20, 2012,
pp. 307-330.
Y. Malitsky, Projected Reflected Gradient Methods for Monotone Variational Inequalities, SIAM J. Control Optim., Vol. 25, 2015, pp. 502-520.
P. Tseng, A Modified Forward-Backward Splitting Method for Maximal Monotone Mappings, SIAM J. Control Optim., Vol. 38, No. 2, 2000, pp. 431-446.
Y. Malitsky, M. K. Tam, A Forward-Backward Splitting Method for Monotone Inclusions without Cocoercivity. SIAM J. Optim., Vol. 30, No. 2, 2020, pp. 1451-1472.
V. Cevher, B. C. Vu, A Reflected Forward-Backward Splitting Method for Monotone Inclusions Involving Lipschitzian Operators. Set-Valued Var. Anal., Vol. 29, 2021, pp. 163-174.
B. T. Polyak, Some Methods of Speeding up the Convergence of Iteration Methods, USSR Comput. Math. Math. Phys., Vol. 4, No. 5, 1964, pp. 1-17.
F. Alvarez, H. Attouch, An Inertial Proximal Method for Monotone Operators Via Discretization of a Nonlinear Oscillator with Damping, Set-Valued Anal., Vol. 9, 2001, pp. 3-11.
Q. L. Dong, D. Jiang, P. Cholamjiak, Y. Shehu, A Strong Convergence Result Involving an Inertial Forward-Backward Algorithm for Monotone Inclusions. J. Fixed Point Theory Appl., Vol. 19, 2007, pp. 3097-3118.
Y. Nesterov, A Method for Solving the Convex Programming Problem with Convergence Rate O(1/k^2 ), Dokl. Akad. Nauk SSSR, Vol. 269, 1983, pp. 543-547.
Q. L. Dong, H. B. Yuan, Y. J. Cho, T. M. Rassias, Modified Inertial Mann Algorithm and Inertial CQ-Algorithm for Nonexpansive Mappings, Optim. Lett., Vol. 12, 2016, pp. 17-33.
D. Lorenz, T. Pock, An Inertial Forward-Backward Algorithm for Monotone Inclusions, J. Math. Imaging Vision, Vol. 51, 2015, pp. 311-325.
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York, 2011.
R. T. Rockafellar, Monotone Operators and the Proximal Point Algorithm, SIAM J. Control Optim., Vol. 14, 1976, pp. 877-898.
M. N. Bui, P. L. Combettes, Multivariate Monotone Inclusions in Saddle Form. Math. Oper. Res., Vol. 47, No. 2, 2022, pp. 1082-1109.
P. L. Combettes, J. C. Pesquet. Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators, Set-Valued Var. Anal., Vol. 20, 2012,
pp. 307-330.
Y. Malitsky, Projected Reflected Gradient Methods for Monotone Variational Inequalities, SIAM J. Control Optim., Vol. 25, 2015, pp. 502-520.
P. Tseng, A Modified Forward-Backward Splitting Method for Maximal Monotone Mappings, SIAM J. Control Optim., Vol. 38, No. 2, 2000, pp. 431-446.
Y. Malitsky, M. K. Tam, A Forward-Backward Splitting Method for Monotone Inclusions without Cocoercivity. SIAM J. Optim., Vol. 30, No. 2, 2020, pp. 1451-1472.
V. Cevher, B. C. Vu, A Reflected Forward-Backward Splitting Method for Monotone Inclusions Involving Lipschitzian Operators. Set-Valued Var. Anal., Vol. 29, 2021, pp. 163-174.
B. T. Polyak, Some Methods of Speeding up the Convergence of Iteration Methods, USSR Comput. Math. Math. Phys., Vol. 4, No. 5, 1964, pp. 1-17.
F. Alvarez, H. Attouch, An Inertial Proximal Method for Monotone Operators Via Discretization of a Nonlinear Oscillator with Damping, Set-Valued Anal., Vol. 9, 2001, pp. 3-11.
Q. L. Dong, D. Jiang, P. Cholamjiak, Y. Shehu, A Strong Convergence Result Involving an Inertial Forward-Backward Algorithm for Monotone Inclusions. J. Fixed Point Theory Appl., Vol. 19, 2007, pp. 3097-3118.
Y. Nesterov, A Method for Solving the Convex Programming Problem with Convergence Rate O(1/k^2 ), Dokl. Akad. Nauk SSSR, Vol. 269, 1983, pp. 543-547.
Q. L. Dong, H. B. Yuan, Y. J. Cho, T. M. Rassias, Modified Inertial Mann Algorithm and Inertial CQ-Algorithm for Nonexpansive Mappings, Optim. Lett., Vol. 12, 2016, pp. 17-33.
D. Lorenz, T. Pock, An Inertial Forward-Backward Algorithm for Monotone Inclusions, J. Math. Imaging Vision, Vol. 51, 2015, pp. 311-325.
H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York, 2011.
R. T. Rockafellar, Monotone Operators and the Proximal Point Algorithm, SIAM J. Control Optim., Vol. 14, 1976, pp. 877-898.