Relaxed forward-backward splitting methods for solving variational inclusions and applications. (English) Zbl 1490.65100
Summary: In this paper, we revisit the modified forward-backward splitting method (MFBSM) for solving a variational inclusion problem of the sum of two operators in Hilbert spaces. First, we introduce a relaxed version of the method (MFBSM) where it can be implemented more easily without the prior knowledge of the Lipschitz constant of component operators. The algorithm uses variable step-sizes which are updated at each iteration by a simple computation. Second, we establish the convergence and the linear rate of convergence of the proposed algorithm. Third, we propose and analyze the convergence of another relaxed algorithm which is a combination between the first one with the inertial method. Finally, we give several numerical experiments to illustrate the convergence of some new algorithms and also to compare them with others.
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
Keywords:
variational inclusion; modified forward-backward splitting method; inertial method; signal recovery; convergence rateReferences:
| [1] | Alvarez, F.; Attouch, H., An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9, 3-11 (2001) · Zbl 0991.65056 · doi:10.1023/A:1011253113155 |
| [2] | Bauschke, HH; Combettes, PL, Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2011), New York: Springer, New York · Zbl 1218.47001 · doi:10.1007/978-1-4419-9467-7 |
| [3] | Davis, D.; Yin, WT, A three-operator splitting scheme and its optimization applications, Set Valued Var. Anal., 25, 829-858 (2017) · Zbl 1464.47041 · doi:10.1007/s11228-017-0421-z |
| [4] | Facchinei, F.; Pang, JS, Finite-Dimensional Variational Inequalities and Complementarity Problems (2003), Berlin: Springer, Berlin · Zbl 1062.90001 |
| [5] | Hartman, P.; Stampacchia, G., On some non-linear elliptic differential-functional equations, Acta Math., 115, 271-310 (1966) · Zbl 0142.38102 · doi:10.1007/BF02392210 |
| [6] | Hieu, DV; Anh, PK; Muu, LD, Modified forward-backward splitting method for variational inclusions, 4OR, 19, 127-151 (2021) · Zbl 1472.65065 · doi:10.1007/s10288-020-00440-3 |
| [7] | Hieu, DV; Anh, PK; Muu, LD; Strodiot, JJ, Iterative regularization methods with new stepsize rules for solving variational inclusions, J. Appl. Math. Comput. (2021) · Zbl 1487.65075 · doi:10.1007/s12190-021-01534-9 |
| [8] | Hieu, DV; Cho, YJ; Xiao, Y-B; Kumam, P., Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math. (2020) · Zbl 07425500 · doi:10.1007/s10013-020-00447-7 |
| [9] | Hieu, DV; Reich, S.; Anh, PK; Ha, NH, A new proximal-like algorithm for solving split variational inclusion problems, Numer. Algorithm (2021) · Zbl 1480.65144 · doi:10.1007/s11075-021-01135-4 |
| [10] | Hieu, DV; Vy, LV; Quy, PK, Three-operator splitting algorithm for a class of variational inclusion problems, Bull. Iran. Math Soc., 46, 1055-1071 (2020) · Zbl 1440.65060 · doi:10.1007/s41980-019-00312-5 |
| [11] | Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), New York: Academic Press, New York · Zbl 0457.35001 |
| [12] | Konnov, IV, Combined Relaxation Methods for Variational Inequalities (2000), Berlin: Springer, Berlin · Zbl 0982.49009 |
| [13] | Lions, PL; Mercier, B., Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16, 964-979 (1979) · Zbl 0426.65050 · doi:10.1137/0716071 |
| [14] | Malitsky, Y.; Tam, MK, A forward-backward splitting method for monotone inclusions without cocoercivity, SIAM J. Optim., 30, 1451-1472 (2020) · Zbl 1445.47041 · doi:10.1137/18M1207260 |
| [15] | Ortega, JM; Rheinboldt, WC, Iterative Solution of Nonlinear Equations in Several Variables (1970), New York: Academic Press, New York · Zbl 0241.65046 |
| [16] | Passty, GB, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72, 383-390 (1979) · Zbl 0428.47039 · doi:10.1016/0022-247X(79)90234-8 |
| [17] | Polyak, BT, Some methods of speeding up the convergence of iterative methods, Zh. Vychisl. Mat. Mat. Fiz., 4, 1-17 (1964) · Zbl 0147.35301 |
| [18] | Ryu, EK; Vu, BC, Finding the Forward-Douglas-Rachford-Forward method, J. Optim. Theory Appl., 184, 858-876 (2020) · Zbl 1513.47112 · doi:10.1007/s10957-019-01601-z |
| [19] | Tseng, P., A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38, 431-446 (2000) · Zbl 0997.90062 · doi:10.1137/S0363012998338806 |
| [20] | Zong, C.; Tang, Y.; Cho, YJ, Convergence analysis of an inexact three-operator splitting algorithm, Symmetry (2018) · Zbl 1423.47017 · doi:10.3390/sym10110563 |
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