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Inertial forward-backward splitting method in Banach spaces with application to compressed sensing. (English) Zbl 07088749
Summary: We propose a Halpern-type forward-backward splitting with inertial extrapolation step for finding a zero of the sum of accretive operators in Banach spaces. Strong convergence of the sequence of iterates generated by the method proposed is obtained under mild assumptions. We give some numerical results in compressed sensing to validate the theoretical analysis results. Our result is one of the few available inertial-type methods for zeros of the sum of accretive operators in Banach spaces.

##### MSC:
 47H05 Monotone operators and generalizations 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators
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##### References:
 [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 (2001), 3-11 [2] Attouch, H.; Cabot, A., Convergence of a relaxed inertial forward-backward algorithm for structured monotone inclusions, Available at https://hal.archives-ouvertes.fr/hal-01782016 (2018), Hal ID: 01782016, 35 pages [3] Baillon, J.-B.; Haddad, G., Quelques propriétés des opérateurs angle-bornés et $$n$$-cycliquement monotones, Isr. J. Math. 26 French (1977), 137-150 [4] Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci. 2 (2009), 183-202 [5] Bertsekas, D. P.; Tsitsiklis, J. N., Parallel and Distributed Computation: Numerical Methods, Athena Scientific, Belmont (2014) [6] Boţ, R. I.; Csetnek, E. R., An inertial alternating direction method of multipliers, Minimax Theory Appl. 1 (2016), 29-49 [7] Boţ, R. I.; Csetnek, E. R.; Hendrich, C., Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput. 256 (2015), 472-487 [8] Brézis, H.; Lions, P.-L., Produits infinis de résolvantes, Isr. J. Math. 29 (1978), 329-345 French [9] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20 (2004), 103-120 [10] Chen, C.; Chan, R. H.; Ma, S.; Yang, J., Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM J. Imaging Sci. 8 (2015), 2239-2267 [11] Chen, G. H.-G.; Rockafellar, R. T., Convergence rates in forward-backward splitting, SIAM J. Optim. 7 (1997), 421-444 [12] Chidume, C., Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics 1965, Springer, Berlin (2009) [13] Cholamjiak, P., A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces, Numer. Algorithms 71 (2016), 915-932 [14] Cholamjiak, P.; Cholamjiak, W.; Suantai, S., A modified regularization method for finding zeros of monotone operators in Hilbert spaces, J. Inequal. Appl. 2015 (2015), Article ID 220, 10 pages [15] Cholamjiak, W.; Cholamjiak, P.; Suantai, S., An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces, J. Fixed Point Theory Appl. 20 (2018), Article ID 42, 17 pages [16] Cioranescu, I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications 62, Kluwer Academic Publishers, Dordrecht (1990) [17] Combettes, P. L., Iterative construction of the resolvent of a sum of maximal monotone operators, J. Convex Anal. 16 (2009), 727-748 [18] Combettes, P. L.; Wajs, V. R., Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005), 1168-1200 [19] Dong, Q.; Jiang, D.; Cholamjiak, P.; Shehu, Y., A strong convergence result involving an inertial forward-backward algorithm for monotone inclusions, J. Fixed Point Theory Appl. 19 (2017), 3097-3118 [20] Dunn, J. C., Convexity, monotonicity, and gradient processes in Hilbert space, J. Math. Anal. Appl. 53 (1976), 145-158 [21] Güler, O., On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29 (1991), 403-419 [22] Halpern, B., Fixed points of nonexpanding maps, Bull. Am. Math. Soc. 73 (1967), 957-961 [23] Haugazeau, Y., Sur la minimisation des formes quadratiques avec contraintes, C. R. Acad. Sci., Paris, Sér. A 264 (1967), 322-324 French [24] Kazarinoff, N. D., Analytic Inequalities, Holt, Rinehart and Winston, New York (1961) [25] Lions, P.-L.; Mercier, B., Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979), 964-979 [26] López, G.; Martín-Márquez, V.; Wang, F.; Xu, H.-K., Forward-backward splitting methods for accretive operators in Banach spaces, Abstr. Appl. Anal. 2012 (2012), Article ID 109236, 25 pages [27] Lorenz, D. A.; Pock, T., An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis. 51 (2015), 311-325 [28] Maingé, P.-E., Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 325 (2007), 469-479 [29] Martinet, B., Régularisation d’inéquations variationnelles par approximations successives, Rev. Franç. Inform. Rech. Opér. 4 (1970), 154-158 French [30] Moudafi, A.; Oliny, M., Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math. 155 (2003), 447-454 [31] Passty, G. B., Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72 (1979), 383-390 [32] Pesquet, J.-C.; Pustelnik, N., A parallel inertial proximal optimization method, Pac. J. Optim. 8 (2012), 273-306 [33] Polyak, B. T., Some methods of speeding up the convergence of iterative methods, U.S.S.R. Comput. Math. Math. Phys. 4 (1967), 1-17 translation from Zh. Vychisl. Mat. Mat. Fiz. 4 1964 791-803 [34] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287-292 [35] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), 877-898 [36] Shehu, Y., Iterative approximations for zeros of sum of accretive operators in Banach spaces, J. Funct. Spaces 2016 (2016), Article ID 5973468, 9 pages [37] Shehu, Y.; Cai, G., Strong convergence result of forward-backward splitting methods for accretive operators in Banach spaces with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 112 (2018), 71-87 [38] Suantai, S.; Pholasa, N.; Cholamjiak, P., The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J. Ind. Manag. Optim. 14 (2018), 1595-1615 [39] Sunthrayuth, P.; Cholamjiak, P., Iterative methods for solving quasi-variational inclusion and fixed point problem in $$q$$-uniformly smooth Banach spaces, Numer. Algorithms 78 (2018), 1019-1044 [40] Tseng, P., A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38 (2000), 431-446 [41] Wei, L.; Agarwal, R. P., A new iterative algorithm for the sum of infinite $$m$$-accretive mappings and infinite $$\mu_i$$-inversely strongly accretive mappings and its applications to integro-differential systems, Fixed Point Theory Appl. 2016 (2016), Article ID 7, 22 pages [42] Xu, H.-K., Inequalities in Banach spaces with applications, Nonlinear Anal., Theory Methods Appl. 16 (1991), 1127-1138
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