Abstract
In this paper, we consider system of variational inclusions and its several spacial cases, namely, alternating point problems, system of variational inequalities, etc., in the setting of Hadamard manifolds. We propose an iterative algorithm for solving system of variational inclusions and study its convergence analysis. Several special cases of the proposed algorithm and convergence result are also presented. We present application to constraints minimization problems for bifunctions in the setting of Hadamard manifolds. At the end, we illustrate proposed algorithms and convergence analysis by a numerical example. The algorithms and convergence results of this paper either improve or extend known algorithms and convergence results from linear structure to Hadamard manifolds.
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Al-Homidan S, Ansari Q H, Babu F. Halpern and Mann type algorithms for fixed points and inclusion problems on Hadamard manifolds. Numer Funct Anal Optim, 2019, 40(6): 621–653
Ansari Q H, Babu F. Proximal point algorithm for inclusion problems in Hadamard manifolds with applications. Optim Lett, 2021, 15(3): 901–921
Ansari Q H, Babu F. Existence and boundedness of solutions to inclusion problems for maximal monotone vector fields in Hadamard manifolds. Optim Lett, 2020, 14(3): 711–727
Ansari Q H, Babu F, Li X B. Variational inclusion problems in Hadamard manifolds. J Nonlinear Convex Anal, 2018, 19(2): 219–237
Ariza-Ruiz D, Lopez-Acedo G, Nicolae A, The asymptotic behavior of the composition of firmly nonexpansive mappings. J Optim Theory Appl, 2015, 167: 409–429
Bačák M, Computing medians and means in Hadamard spaces. SIAM J Optim, 2014, 24: 1542–1566
Bauschke H H, Combettes P L, Reich S, The asymptotic behavior of the composition of two resolvents. Nonlinear Anal, 2005, 60: 283–301
Browder F E, Petryshyn W V, Construction of fixed points of nonlinear in Hilbert space. J Math Anal Appl, 1967, 20: 197–228
Ceng L C, Wang C Y, Yao J C, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math Meth Oper Res, 2008, 67: 375–390
da Cruz Neto J X, Ferreira O P, Lucambio P R. Monotone point-to-set vector fields. Balk J Geom Appl, 2000, 5(1): 69–79
da Cruz Neto J X, Ferreira O P, Lucambio P R, Németh S Z, Convex-and monotone-transformable mathematical programming problems and a proximal-like point method. J Global Optim, 2006, 35: 53–69
do Carmo M P. Riemannian Geometry. Boston, Basel, Berlin: Birkhäuser, 1992
Ferreira O P, Oliveira P R, Proximal point algorithm on Riemannian manifolds. Optimization, 2002, 51: 257–270
Iusem A N, An iterative algorithm for the variational inequality problem. Comput Appl Math, 1994, 13: 103–114
Li C, López G, Martín-Márquez V, Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J Lond Math Soc, 2009 79: 663–683
Li C, López G, Martíín-Máarquez V. Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J Math, 2010, 14(2): 541–559
Li C, López G, Martín-Márquez V, Resolvent of set-valued monotone vector fields in Hadamard manifolds. Set-valued Anal, 2011 19: 361–383
Liu L S, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl, 1995, 194: 114–125
Moudafi A. A three-operator splitting algorithm for null-point problems. Fixed Point Theory, 2020, 21(2): 685–692
Németh S Z. Monotonicity of the complementary vector field of a nonexpansive map. Acta Math Hungar, 1999, 84(3): 18–197
Németh S Z, Monotone vector fields. Publ Math Debrecen, 1999, 54: 437–449
Passty G B, Ergodic convergence to a zero of the sum of maximal monotone operators. J Math Anal Appl, 1978, 7: 591–597
Rapcsák T. Smooth Nonlinear Optimization in ℝn. Dordrecht: Kluwer Academic Publishers, 1997
Reich S, Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl, 1980, 75: 287–292
Sahu D R. Altering points and applications. Nonlinear Stud, 2014, 21(2): 349–365
Sahu D R, Ansari Q H, Yao J C, The prox-Tikhonov-like forward-backward method and application. Taiwanese J Math, 2015 19: 481–503
Sahu D R, Babu F, Sharma S. The S-iterative techniques on Hadamard manifolds and applications. J Appl Numer Optim, 2020, 2(3): 353–371
Sakai T. Riemannian Geometry. Providence, RI: Amer Math Soc, 1996
Sakurai K, Jimba T, Iiduka H. Iterative methods for parallel convex optimization with fixed point constraints. J Nonlinear Var Anal, 2019, 3(2): 115–126
Tuyen T M, Quy T X, Trang N M. A parallel iterative method for solving a class of variational inequalities in Hilbert spaces. J Nonlinear Var Anal, 2020, 4(3): 357–376
Udriste C. Convex Functions and Optimization Methods on Riemannian Manifolds. Dordrecht: Kluwer Academic Publishers, 1994
Walter R, On the metric projections onto convex sets in Riemannian spaces. Arch Math, 1974, 25: 91–98
Zhang C, Chen J, The subgradient extragradient-type algorithms for solving a class of monotone variational inclusion problems. J Appl Numer Optim, 2020, 2: 3
Zhao X P, Sahu D R, Wen C F. Iterative methods for system of variational inclusions involving accretive operators and applications. Fixed Point Theory, 2018, 19(2): 801–822
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Ansari, Q.H., Babu, F. & Sahu, D.R. Iterative Algorithms for System of Variational Inclusions in Hadamard Manifolds. Acta Math Sci 42, 1333–1356 (2022). https://doi.org/10.1007/s10473-022-0405-4
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DOI: https://doi.org/10.1007/s10473-022-0405-4
Key words
- System of variational inclusions
- altering point problems
- monotone vector fields
- strictly pseudocontractive mappings
- Hadamard manifolds