ITERATIVE ALGORITHMS FOR SYSTEM OF VARIATIONAL INCLUSIONS IN HADAMARD MANIFOLDS

doi:10.1007/s10473-022-0405-4

Abstract

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.

Key words: System of variational inclusions, altering point problems, monotone vector fields, strictly pseudocontractive mappings, Hadamard manifolds

CLC Number:

49J53

Qamrul Hasan ANSARI, Feeroz BABU, D. R. SAHU. ITERATIVE ALGORITHMS FOR SYSTEM OF VARIATIONAL INCLUSIONS IN HADAMARD MANIFOLDS[J].Acta mathematica scientia,Series B, 2022, 42(4): 1333-1356.

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References

[1] 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
[2] Ansari Q H, Babu F. Proximal point algorithm for inclusion problems in Hadamard manifolds with applications. Optim Lett, 2021, 15(3):901-921
[3] 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
[4] Ansari Q H, Babu F, Li X B. Variational inclusion problems in Hadamard manifolds. J Nonlinear Convex Anal, 2018, 19(2):219-237
[5] 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
[6] Bačák M, Computing medians and means in Hadamard spaces. SIAM J Optim, 2014, 24:1542-1566
[7] Bauschke H H, Combettes P L, Reich S, The asymptotic behavior of the composition of two resolvents. Nonlinear Anal, 2005, 60:283-301
[8] Browder F E, Petryshyn W V, Construction of fixed points of nonlinear in Hilbert space. J Math Anal Appl, 1967, 20:197-228
[9] 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
[10] 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
[11] 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
[12] do Carmo M P. Riemannian Geometry. Boston, Basel, Berlin:Birkhäuser, 1992
[13] Ferreira O P, Oliveira P R, Proximal point algorithm on Riemannian manifolds. Optimization, 2002, 51:257-270
[14] Iusem A N, An iterative algorithm for the variational inequality problem. Comput Appl Math, 1994, 13:103-114
[15] Li C, López G, Martín-Márquez V, Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J Lond Math Soc, 200979:663-683
[16] Li C, López G, Martíín-Máarquez V. Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J Math, 2010, 14(2):541-559
[17] Li C, López G, Martín-Márquez V, Resolvent of set-valued monotone vector fields in Hadamard manifolds. Set-valued Anal, 201119:361-383
[18] 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
[19] Moudafi A. A three-operator splitting algorithm for null-point problems. Fixed Point Theory, 2020, 21(2):685-692
[20] Németh S Z. Monotonicity of the complementary vector field of a nonexpansive map. Acta Math Hungar, 1999, 84(3):18-197
[21] Németh S Z, Monotone vector fields. Publ Math Debrecen, 1999, 54:437-449
[22] Passty G B, Ergodic convergence to a zero of the sum of maximal monotone operators. J Math Anal Appl, 1978, 7:591-597
[23] Rapcsák T. Smooth Nonlinear Optimization in $\mathbb{R}$ⁿ. Dordrecht:Kluwer Academic Publishers, 1997
[24] Reich S, Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl, 1980, 75:287-292
[25] Sahu D R. Altering points and applications. Nonlinear Stud, 2014, 21(2):349-365
[26] Sahu D R, Ansari Q H, Yao J C, The prox-Tikhonov-like forward-backward method and application. Taiwanese J Math, 201519:481-503
[27] 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
[28] Sakai T. Riemannian Geometry. Providence, RI:Amer Math Soc, 1996
[29] 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
[30] 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
[31] Udriste C. Convex Functions and Optimization Methods on Riemannian Manifolds. Dordrecht:Kluwer Academic Publishers, 1994
[32] Walter R, On the metric projections onto convex sets in Riemannian spaces. Arch Math, 1974, 25:91-98
[33] 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
[34] 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