Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

Hammed Anuoluwapo Abbas; Kazeem Aremu; Olawale Oyewole; Akindele Mebawondu; Ojen Narain

doi:10.33993/jnaat522-1351

Authors

Hammed Anuoluwapo Abbas Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Science University, Ga-Rankuwa, South Africa https://orcid.org/0000-0002-4236-3278
Kazeem Aremu Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Science University, Ga-Rankuwa, South Africa
Olawale Oyewole School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
Akindele Mebawondu Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Nigeria
Ojen Narain School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

DOI:

https://doi.org/10.33993/jnaat522-1351

Keywords:

Nonexpansive mapping, minimization problem, inertial method, forward-backward splitting method, fixed point problem

Abstract views: 216

Abstract

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

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References

H.A. Abass, K.O. Aremu, L. O. Jolaoso and O.T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problem, J. Nonlinear Funct. Anal., 2020 (2020), Article ID 6, https://doi.org/10.23952/jnfa.2020.6 DOI: https://doi.org/10.23952/jnfa.2020.6

H.A. Abass, C. Izuchukwu, O.T. Mewomo and Q.L. Dong, Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach spaces, Fixed Point Theory, vol. 21, no. 2, (2020) pp. 397–412, https://doi.org/10.24193/fpt-ro.2020.2.28. DOI: https://doi.org/10.24193/fpt-ro.2020.2.28

H.A. Abass, C. Izuchukwu, O.T. Mewomo and F.U. Ogbuisi, An iterative method for solutions of finite families of split minimization problems and fixed point problems, Novi Sad Journal of Mathematics, 49 (2019) no. 1, pp. 117–136, https://doi.org/10.30755/NSJOM.07925. DOI: https://doi.org/10.30755/NSJOM.07925

H.A. Abass and L.O. Jolaoso, An inertial generalized viscosity approximation method for solving multiple-sets split feasibility problem and common fixed point of strictly pseudo-nonspreading mappings, Axioms, 10, no. 1, (2021), https://doi.org/10.3390/axioms10010001. DOI: https://doi.org/10.3390/axioms10010001

M. Abbas, M. Alshahrani, Q.H. Ansari, O.S. Iyiola and Y. Shehu, Iterative methods for solving proximal split minimization problems, Numer. Algorithms, 78 (2018) no. 1, pp. 193–215, https://doi.org/10.1007/s11075-017-0372-3. DOI: https://doi.org/10.1007/s11075-017-0372-3

F. Alvarez and H. Attouch, An Inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9, (2001), pp. 3–11, https://doi.org/10.1023/A:1011253113155. DOI: https://doi.org/10.1023/A:1011253113155

Q.H. Ansari and A. Rehan, Split feasibility and fixed point problems. In Nonlinear Analysis: Approximation Theory, Optimization and Application, ed. Q.H. Ansari, 281–322. New York: Springer, 2014, https://doi.org/10.1007/978-81-322-1883-8_9. DOI: https://doi.org/10.1007/978-81-322-1883-8_9

K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal., 67, (2007), pp. 2350–2360, https://doi.org/10.1016/j.na.2006.08.032. DOI: https://doi.org/10.1016/j.na.2006.08.032

H.H. Bauschke and P.L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 408, New York: Springer, 2011, https://doi.org/10.1007/978-1-4419-9467-7. DOI: https://doi.org/10.1007/978-1-4419-9467-7

J.Y. Bello Crus and T.T. Nghia, On the convergence of the forward-backward splitting method with line searches, Optim. Methods Softw., 31 (2016) no. 6, pp. 1209–1238, https://doi.org/10.1080/10556788.2016.1214959. DOI: https://doi.org/10.1080/10556788.2016.1214959

C. Bryne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Problems, 18 (2002) no. 2, pp. 441-–453, https://iopscience.iop.org/article/10.1088/0266-5611/18/2/310/meta. DOI: https://doi.org/10.1088/0266-5611/18/2/310

C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), pp. 103—120, https://iopscience.iop.org/article/10.1088/0266-5611/20/1/006/meta. DOI: https://doi.org/10.1088/0266-5611/20/1/006

C. Bryne, Y. Censor, A. Gibali and S. Reich, Weak and strong convergence of algorithms of the split common null point problem, J. Nonlinear Convex Analysis, 13, (2012), pp. 759–775, e2ceeb131729e8401319e4b64c69c2021fa13d84.

R.S. Burachik and A.N. Iusem, Set-valued mappings and enlargements of monotone operators, 8, Boston: Springer Science Business Media, 2007.

Y. Censor, T. Bortfield, B. Martin and A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), pp. 2353–2365, https://doi.org/10.1088/0031-9155/51/10/001. DOI: https://doi.org/10.1088/0031-9155/51/10/001

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in product space, Numer. Algorithms, 8, (1994), pp. 221–239, https://doi.org/10.1007/BF02142692. DOI: https://doi.org/10.1007/BF02142692

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), pp. 2071–2084, https://doi.org/10.1088/0266-5611/21/6/017. DOI: https://doi.org/10.1088/0266-5611/21/6/017

S.S. Chang, J.C. Yao, L. Wang, M. Liu and L. Zhao, On the inertial forward-backward splitting technique for solving a system of inclusion problems in Hilbert spaces,Optimization, (2020), pp. 1–15, https://doi.org/10.1080/02331934.2020.1786567. DOI: https://doi.org/10.1080/02331934.2020.1786567

C.E Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer Verlag Series, Lecture Notes in Mathematics, ISBN 978-1-84882-189-7, 2009, https://doi.org/10.1007/978-1-84882-190-3. DOI: https://doi.org/10.1007/978-1-84882-190-3

W. Cholamjiak, P. Cholamjiak and S. Suantai, An inertial forward-backward splitting method for solving inclusion problems in Hilbert space, J. Fixed Point Theor. Appl., (2018), pp. 20–42, https://doi.org/10.1007/s11784-018-0526-5. DOI: https://doi.org/10.1007/s11784-018-0526-5

P.L. Combettes and J.C. Pesquet, Proximal splitting methods in signal processing in H.H. Bauschke, R. Burachik, P.L. Combettes, V. Elser, D.R. Wolkowicz, H. (eds) Fixed Point Algorithms for inverse problems in science and engineering, 185212, Springer, New York, 2011, https://doi.org/10.1007/978-1-4419-9569-8_10. DOI: https://doi.org/10.1007/978-1-4419-9569-8_10

K. Goebel and S. Reich, Uniform convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984, http://www.gbv.de/dms/hbz/toc/ht002523053.pdf.

F.O. Isiogugu and M.O. Osilike, Convergence theorems for new classes of multi-valued hemicontractive-type mappings, Fixed Point Theory and Applications, 93 (2014), https://doi.org/10.1186/1687-1812-2014-93. DOI: https://doi.org/10.1186/1687-1812-2014-93

L.O. Jolaoso, H.A. Abass and O.T. Mewomo, A Viscosity-Proximal Gradient Method with Inertial Extrapolation for Solving Minimization Problem in Hilbert Space, Arch. Math. (BRNO), Tomus 55 (2019), pp. 167–194, https://dml.cz/handle/10338.dmlcz/147824. DOI: https://doi.org/10.5817/AM2019-3-167

C. Kanzow and Y. Shehu, Strong convergence of a double-type method for monotone variational inequalities in Hilbert spaces, J. Fixed Point Theory Appl., 20 (2018) no. 1, Art. 51, pp. 1–24, https://doi.org/10.1007/s11784-018-0531-8. DOI: https://doi.org/10.1007/s11784-018-0531-8

Y. Kimura and S. Saejung, Strong convergence for a common fixed points of two different generalizations of cutter operators, Linear Nonlinear Anal., 1 (2015), pp. 53–65.

K. Kunrada and P. Cholamjiak, Convergence of the forward-backward splitting algorithms via linesearches in Hilbert spaces, Applicable Analysis, (2021), pp. 1–20, https://doi.org/10.1080/00036811.2021.1986021. DOI: https://doi.org/10.1080/00036811.2021.1986021

L.J. Lin and W. Takahashi, A general iterative method for hierachical varaitional inequality problems in Hilbert spaces and applications, Positivity, (2012), 16 (2012) no. 3, pp. 429–453, https://doi.org/10.1007/s11117-012-0161-0. DOI: https://doi.org/10.1007/s11117-012-0161-0

D.A. Lorenz and T. Pock, An inertial forward-backward splitting algorithm fpr monotone inclusions, J. Math. Imaging Vis., 51 (2015), pp. 311–325, https://doi.org/10.1007/s10851-014-0523-2. DOI: https://doi.org/10.1007/s10851-014-0523-2

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), pp. 275–288, https://doi.org/10.1007/s10957-011-9814. DOI: https://doi.org/10.1007/s10957-011-9814-6

P.E. Mainge, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set-Valued Anal., 15 (2007), pp. 67–79, https://doi.org/10.1007/s11228-006-0027-3. DOI: https://doi.org/10.1007/s11228-006-0027-3

W. Phuengrattana and J. Tiammee, Proximal point algorithms for finding common fixed points of a finite family of quasi-nonexpansive multi-valued mappings in real Hilbert spaces, J. Fixed Point Theory Appl., 20 (2018), pp. 1–14, https://doi.org/10.1007/s11784-018-0590-x. DOI: https://doi.org/10.1007/s11784-018-0590-x

B.T. Polyak, Some methods of speeding up the convergence of iterates methods, U.S.S.R Comput. Math. Phys. 4 (1964) no. 5, pp. 1–17, https://doi.org/10.1016/0041-5553(64)90137-5. DOI: https://doi.org/10.1016/0041-5553(64)90137-5

Y. Shehu and O.S. Iyiola, On a modified extragradient method for variational inequality problem with application to industrial electricity production, J. Ind. Manag. Optim., 15 (2019) no. 1, pp. 319–342, https://doi.org/10.3934/jimo.2018045. DOI: https://doi.org/10.3934/jimo.2018045

Y. Shehu and F.U. Ogbuisi, An iterative method for solving split monotone variational inclusion and fixed point problems, Revista de la Real Academia de Ciencias Exactas, Fiscas y Naturales, Serie A Matemaicas, 110 (2016) no. 2, pp. 503–518, https://doi.org/10.1007/s13398-015-0245-3. DOI: https://doi.org/10.1007/s13398-015-0245-3

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

W. Takahashi,Introduction to nonlinear and convex analysis, Yokohama Publisher, Yokohama, 2009.

D.V. Thong and D.V. Hieu, A new approximation method for finding common fixed points of families of demicontractive operators and applications, 20 (2018), pp. 1–27, https://doi.org/10.1007/s11784-018-0551-4. DOI: https://doi.org/10.1007/s11784-018-0551-4

Y. Wang and F. Wang, Strong convergence of the forward-backward splitting method with multiple parameters in Hilbert spaces, Optimization, 67 (2018), no. 4, pp. 493–505, https://doi.org/10.1080/02331934.2017.1411485. DOI: https://doi.org/10.1080/02331934.2017.1411485

H.K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theor. Apl., 150 (2011), pp. 360—378, https://doi.org/10.1007/s10957-011-9837-z. DOI: https://doi.org/10.1007/s10957-011-9837-z

H.Y. Zhou, Convergence theorems of fixed points for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 69 (2008) no. 2, pp. 456–462, https://doi.org/10.1016/j.na.2007.05.032. DOI: https://doi.org/10.1016/j.na.2007.05.032