An inertial Mann algorithm for nonexpansive mappings on Hadamard manifolds

Konrawut Khammahawong; Parin Chaipunya; Poom Kumam; Konrawut Khammahawong; Parin Chaipunya; Poom Kumam

doi:10.3934/math.2023108

AIMS Mathematics

2023, Volume 8, Issue 1: 2093-2116. doi: 10.3934/math.2023108

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An inertial Mann algorithm for nonexpansive mappings on Hadamard manifolds

1.
Applied Mathematics for Science and Engineering Research Unit (AMSERU), Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Pathum Thani 12110, Thailand
2.
NCAO Research Center, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand
3.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTT Fixed Point Research Laboratory, Fixed Point Laboratory, Science Laboratory Building, Departments of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand

Received: 29 June 2022 Revised: 20 September 2022 Accepted: 13 October 2022 Published: 28 October 2022
MSC : 47H05, 47J25

An inertial Mann algorithm will be presented in this article with the purpose of approximating a fixed point of a nonexpansive mapping on a Hadamard manifold. Any sequence that is generated by using the proposed approach, under suitable assumptions, converges to fixed points of nonexpansive mappings. The proposed method is also dedicated to solving inclusion and equilibrium problems. Lastly, we give a number of computational experiments that show how well the inertial Mann algorithm works and how it compares to other methods.
- fixed point problem,
- Hadamard manifold,
- inertial Mann method,
- nonexpansive mapping
Citation: Konrawut Khammahawong, Parin Chaipunya, Poom Kumam. An inertial Mann algorithm for nonexpansive mappings on Hadamard manifolds[J]. AIMS Mathematics, 2023, 8(1): 2093-2116. doi: 10.3934/math.2023108

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Abstract

An inertial Mann algorithm will be presented in this article with the purpose of approximating a fixed point of a nonexpansive mapping on a Hadamard manifold. Any sequence that is generated by using the proposed approach, under suitable assumptions, converges to fixed points of nonexpansive mappings. The proposed method is also dedicated to solving inclusion and equilibrium problems. Lastly, we give a number of computational experiments that show how well the inertial Mann algorithm works and how it compares to other methods.

References

[1]	O. P. Ferreira, L. R. Lucambio Pérez, S. Z. Németh, Singularities of monotone vector fields and an extragradient-type algorithm, J. Glob. Optim., 31 (2005), 133–151. https://doi.org/10.1007/s10898-003-3780-y doi: 10.1007/s10898-003-3780-y
[2]	C. Li, G. López, V. Martín-Márquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc., 79 (2009), 663–683. https://doi.org/10.1112/jlms/jdn087 doi: 10.1112/jlms/jdn087
[3]	T. Sakai, Riemannian geometry, American Mathematical Society, Vol. 149, 1996.
[4]	M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, Vol. 319, 1999. https://doi.org/10.1007/978-3-662-12494-9
[5]	J. X. da Cruz Neto, O. P. Ferreira, L. R. Lucambio Pérez, Monotone point-to-set vector fields, Balkan J. Geom. Appl., 5 (2000), 69–79.
[6]	V. Colao, G. López, G. Marino, V. Martín-Márquez, Equilibrium problems in Hadamard manifolds, J. Math. Anal. Appl., 388 (2012), 61–77. https://doi.org/10.1016/j.jmaa.2011.11.001 doi: 10.1016/j.jmaa.2011.11.001
[7]	X. Wang, G. López, C. Li, J. Yao, Equilibrium problems on Riemannian manifolds with applications, J. Math. Anal. Appl., 473 (2019), 866–891. https://doi.org/10.1016/j.jmaa.2018.12.073 doi: 10.1016/j.jmaa.2018.12.073
[8]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.2307/2032162 doi: 10.2307/2032162
[9]	P. Maingé, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2018), 223–236. https://doi.org/10.1016/j.cam.2007.07.021 doi: 10.1016/j.cam.2007.07.021
[10]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Elsevier, 1964. https://doi.org/10.1016/0041-5553(64)90137-5
[11]	C. Li, G. López, V. Martín-Márquez, Iterative algorithms for nonexpansive mappings on Hadamard manifolds, Taiwan. J. Math., 14 (2010), 541–559. https://doi.org/10.11650/twjm/1500405806 doi: 10.11650/twjm/1500405806
[12]	R. Chugh, M. Kumari, A. Kumar, Two-step iterative procedure for non-expansive mappings on Hadamard manifolds, Commun. Optim. Theory, 2014.
[13]	T. T. Yao, Y. H. Li, Y. S. Zhang, Z. Zhao, A modified Riemannian Halpern algorithm for nonexpansive mappings on Hadamard manifolds, Optimization, 2021. https://doi.org/10.1080/02331934.2021.1914036 doi: 10.1080/02331934.2021.1914036
[14]	D. R. Sahu, F. Babu, S. Sharma, The S-iterative techniques on Hadamard manifolds and applications, J. Appl. Numer. Optim., 2 (2020), 353–371. https://doi.org/10.23952/jano.2.2020.3.06 doi: 10.23952/jano.2.2020.3.06
[15]	S. Chang, J. C. Yao, M. Liu, L. C. Zhao, J. H. Zhu, Shrinking projection algorithm for solving a finite family of quasi-variational inclusion problems in Hadamard manifold, RACSAM, 115 (2021), 166. https://doi.org/10.1007/s13398-021-01105-4 doi: 10.1007/s13398-021-01105-4
[16]	S. Huang, Approximations with weak contractions in Hadamard manifolds, Linear Nonlinear Anal., 1 (2015), 317–328.
[17]	C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Prob., 20 (2004), 103–120. https://doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
[18]	E. Hacıoğlu, F. Gürsoy, S. Maldar, Y. Atalan, G. V. Milovanović, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning, Appl. Numer. Math., 167 (2021), 143–172. https://doi.org/10.1016/j.apnum.2021.04.020 doi: 10.1016/j.apnum.2021.04.020
[19]	C. I. Podilchuk, R. J. Mammone, Image recovery by convex projections using a least-squares constraint, J. Optical Soc. Amer. A, 7 (1990), 517–521. https://doi.org/10.1364/josaa.7.000517 doi: 10.1364/josaa.7.000517
[20]	A. Padcharoen, P. Sukprasert, Nonlinear operators as concerns convex programming and applied to signal processing, Mathematics, 7 (2019), 866. https://doi.org/10.3390/math7090866 doi: 10.3390/math7090866
[21]	S. Al-Homidan, Q. H. Ansari, F. Babu, Halpern- and Mann-type algorithms for fixed points and inclusion problems on Hadamard manifolds, Numer. Funct. Anal. Optim., 40 (2019), 621–653. https://doi.org/10.1080/01630563.2018.1553887 doi: 10.1080/01630563.2018.1553887
[22]	J. Hu, X. Liu, Z. W. Wen, Y. X. Yuan, A brief introduction to manifold optimization, J. Oper. Res. Soc. China, 8 (2020), 199–248. https://doi.org/10.1007/s40305-020-00295-9 doi: 10.1007/s40305-020-00295-9
[23]	C. Li, G. López, V. Martín-Márquez, J. H. Wang, Resolvents of set-valued monotone vector fields in Hadamard manifolds, Set-Valued Anal., 19 (2021), 361–383. https://doi.org/10.1007/s11228-010-0169-1 doi: 10.1007/s11228-010-0169-1
[24]	O. P. Ferreira, M. S. Louzeiro, L. F. Prudente, Gradient method for optimization on Riemannian manifolds with lower bounded curvature, SIAM J. Optim., 29 (2019), 2517–2541. https://doi.org/10.1137/18M1180633 doi: 10.1137/18M1180633
[25]	J. X. Da Cruz Neto, O. P. Ferreira, L. R. Lucambio Pérez, S. Z. Németh, Convex- and monotone-transformable mathematical programming problems and a proximal-like point method, J. Glob. Optim., 35 (2006), 53–69. https://doi.org/10.1007/s10898-005-6741-9 doi: 10.1007/s10898-005-6741-9
[26]	F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2003), 773–782. https://doi.org/10.1137/S1052623403427859 doi: 10.1137/S1052623403427859
[27]	F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3–11. https://doi.org/10.1023/A:1011253113155 doi: 10.1023/A:1011253113155
[28]	D. A. Lorenz, T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51 (2015), 311–325. https://doi.org/10.1007/s10851-014-0523-2 doi: 10.1007/s10851-014-0523-2
[29]	P. Ochs, T. Brox, T. Pock, iPiasco: Inertial proximal algorithm for strongly convex optimization, J. Math. Imaging Vis., 53 (2015), 171–181. https://doi.org/10.1007/s10851-015-0565-0 doi: 10.1007/s10851-015-0565-0
[30]	J. Fan, L. Liu, X. Qin, A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities, Optimization, 69 (2020), 2199–2215. https://doi.org/10.1080/02331934.2019.1625355 doi: 10.1080/02331934.2019.1625355
[31]	N. Van Dung, N. T. Hieu, A new hybrid projection algorithm for equilibrium problems and asymptotically quasi $\phi$-nonexpansive mappings in Banach spaces, RACSAM, 113 (2019), 2017–2035. https://doi.org/10.1007/s13398-018-0595-8 doi: 10.1007/s13398-018-0595-8
[32]	J. Chen, S. Liu, Extragradient-like method for pseudomontone equilibrium problems on Hadamard manifolds, J. Inequal. Appl., 2020 (2020), 205. https://doi.org/10.1186/s13660-020-02473-y doi: 10.1186/s13660-020-02473-y
[33]	B. Tan, Z. Zhou, S. Li, Strong convergence of modified inertial Mann algorithms for nonexpansive mappings, Mathematics, 8 (2020), 462. https://doi.org/10.3390/math8040462 doi: 10.3390/math8040462
[34]	B. Tan, J. Fan, X. Qin, Inertial extragradient algorithms with non-monotonic step sizes for solving variational inequalities and fixed point problems, Adv. Oper. Theory, 6 (2021), 61. https://doi.org/10.1007/s43036-021-00155-0 doi: 10.1007/s43036-021-00155-0
[35]	A. N. Iusem, V. Mohebbi, An extragradient method for vector equilibrium problems on Hadamard manifolds, J. Nonlinear Var. Anal., 5 (2021), 459–476. https://doi.org/10.23952/jnva.5.2021.3.09 doi: 10.23952/jnva.5.2021.3.09
[36]	F. Babu, Iterative methods for certain problems from nonlinear analysis in the setting of manifolds, Aligarh Muslim University, PhD Dissertation, 2019.
[37]	T. Rapcsák, Smooth nonlinear optimization in ${\bf R}^ n$, New York: Springer, 1997. https://doi.org/10.1007/978-1-4615-6357-0
[38]	K. Khammahawong, P. Kumam, P. Chaipunya, Splitting algorithms for equilibrium problems and inclusion problems on Hadamard manifolds, Numer. Funct. Anal. Optim., 42 (2021), 1645–1682. https://doi.org/10.1080/01630563.2021.1933523 doi: 10.1080/01630563.2021.1933523
[39]	P. Chaipunya, K. Khammahawong, P. Kumam, Iterative algorithm for singularities of inclusion problems in Hadamard manifolds, J. Inequal. Appl., 2021 (2021), 147. https://doi.org/10.1186/s13660-021-02676-x doi: 10.1186/s13660-021-02676-x
[40]	K. Khammahawong, P. Chaipunya, P. Kumam, Iterative algorithms for monotone variational inequality and fixed point problems on Hadamard manifolds, Adv. Oper. Theory, 7 (2022), 43. https://doi.org/10.1007/s43036-022-00207-z doi: 10.1007/s43036-022-00207-z
[41]	Q. L. Dong, H. B. Yuan, Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping, Fixed Point Theory Appl., 2015 (2015), 125. https://doi.org/10.1186/s13663-015-0374-6 doi: 10.1186/s13663-015-0374-6
[42]	Q. L. Dong, H. B. Yuan, Y. J. Cho, Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87–102. https://doi.org/10.1007/s11590-016-1102-9 doi: 10.1007/s11590-016-1102-9
[43]	H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-9467-7
[44]	G. Kassay, V. D. Rădulescu, Equilibrium problems and applications, London: Elsevier/Academic Press, 2019.

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