Convergence analysis for variational inclusion problems equilibrium problems and fixed point in Hadamard manifolds. (English) Zbl 07376437
The authors consider the problem of finding
\[
x^* \in \mathrm{Fix} (S) \cap EP(F) \cap \bigcap_{j=1}^n (A_j+B)^{-1} (0)\tag{1}
\]
in a Hadamard manifold, where \(Fix(S)\) is the set of fixed points of a quasi-nonexpansive mapping \(S\), \(EP(F)\) is the set of equilibrium points of an equilibrium function \(F\), \(B\) is a set-valued maximal monotone mapping, \(A_j\), \(j=1, 2, \dots, n\), are single-valued continuous and monotone mappings and \(\bigcap_{j=1}^n (A_j +B)^{-1} (0)\) is the set of common singularities of a system of quasi-variational inclusion problems, which is motivated and inspired by the works given by V. Colao et al. [J. Math. Anal. Appl. 388, No. 1, 61–77 (2012; Zbl 1273.49015)] and by S. Al-Homidan et al. [Numer. Funct. Anal. Optim. 40, No. 6, 621–653 (2019; Zbl 1447.49032)]. The authors propose a new algorithm to find a common solution of the problem (1) and to study the minimization problem and saddle point problem in Hadamard manifolds.
The results are useful for researchers on quasi-variational problems in Hadamard manifolds and related mathematical inequalities.
The results are useful for researchers on quasi-variational problems in Hadamard manifolds and related mathematical inequalities.
Reviewer: Choonkil Park (Seoul)
MSC:
| 47H10 | Fixed-point theorems |
| 47H05 | Monotone operators and generalizations |
| 47J25 | Iterative procedures involving nonlinear operators |
| 58A05 | Differentiable manifolds, foundations |
| 58C30 | Fixed-point theorems on manifolds |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
Hadamard manifold; equilibrium problem; maximal monotone vector field; quasi-variational inclusion problem; quasi-nonexpansive mappingReferences:
| [1] | Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim, 14, 5, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056 |
| [2] | Fan, K., A Minimax Inequality and Applications, 103-113 (1972), New York: Academic Press, New York · Zbl 0302.49019 |
| [3] | Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math Student, 63, 1-4, 123-145 (1994) · Zbl 0888.49007 |
| [4] | Chang, S. S., Set-valued variational inclusions in banach spaces, J. Math. Anal. Appl, 248, 2, 438-454 (2000) · Zbl 1031.49018 · doi:10.1006/jmaa.2000.6919 |
| [5] | Chang, S. S., Existence and approximation of solutions for set-valued variational inclusions in Banach spaces, Nonlinear Anal, 47, 1, 583-594 (2001) · Zbl 1042.49510 · doi:10.1016/S0362-546X(01)00203-6 |
| [6] | Chang, S. S.; Cho, Y. J.; Lee, B. S.; Jung, I. H., Generalized set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl, 246, 2, 409-422 (2000) · Zbl 1031.49017 · doi:10.1006/jmaa.2000.6795 |
| [7] | Khammahawong, K., Kumam, P., Chaipunya, P. (2019). Splitting algorithms of common solutions between equilibrium and inclusion problems on Hadamard manifolds, arXiv:1907.00364v1 [math.FA] 30 Jun 2019. · Zbl 1433.90170 |
| [8] | Sahu, D. R.; Ansari, Q. H.; Yao, J. C., The Prox-Tikhonov forward-backward method and applications, Taiwanese J. Math, 19, 2, 481-503 (2015) · Zbl 1357.47077 · doi:10.11650/tjm.19.2015.4972 |
| [9] | Takahashi, S.; Takahashi, W.; Toyoda, M., Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147, 1, 27-41 (2010) · Zbl 1208.47071 · doi:10.1007/s10957-010-9713-2 |
| [10] | Chang, S. S.; Lee, J. H. W.; Chan, C. K., Algorithms of common solutions to quasi variational inclusion and fixed point problems, Appl. Math. Mech. Engl. Ed, 29, 5, 571-581 (2008) · Zbl 1196.47047 |
| [11] | Li, C.; Yao, J. C., Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm, SIAM J. Control Optim, 50, 4, 2486-2514 (2012) · Zbl 1257.49011 · doi:10.1137/110834962 |
| [12] | Konnov, I., V. Equilibrium models, vol. 210 of Mathematics in Science and Engineering, and variational inequalities (2007), Amsterdam: Elsevier B. V, Amsterdam · Zbl 1140.91056 |
| [13] | Muu, L. D.; Oettli, W., Convergence of an adaptive penalty scheme for find- ing constrained equilibria, Nonlinear Anal, 18, 12, 1159-1166 (1992) · Zbl 0773.90092 · doi:10.1016/0362-546X(92)90159-C |
| [14] | Li, C.; López, G.; Martín-Márquez, V., Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. Lond. Math. Soc, 79, 3, 663-683 (2009) · Zbl 1171.58001 · doi:10.1112/jlms/jdn087 |
| [15] | Li, C.; López, G.; Martín-Márquez, V., Iterative algorithms for nonexpansive mappings on Hadamard manifolds, Taiwan. J. Math, 14, 2, 541-559 (2010) · Zbl 1217.47118 |
| [16] | Ansari, Q. H.; Babu, F.; Li, X., Variational inclusion problems in Hadamard manifolds, J. Nonlinear Convex Anal, 19, 2, 219-237 (2018) · Zbl 1395.49013 |
| [17] | Suliman, A.-H.; Ansari, Q. H.; Babu, F., Halpern- and Mann-type algorithms for fixed points and inclusion problems on Hadamard Manifolds, Numer. Funct. Anal. Opt, 40, 6, 621-653 (2019) · Zbl 1447.49032 |
| [18] | Iusem, A. N., An iterative algorithm for the variational inequality problem, Comput. Appl. Math, 13, 103-114 (1994) · Zbl 0811.65049 |
| [19] | Li, C.; López, G.; Martín-Márquez, V.; Wang, J.-H., Resolvents of set-valued monotone vector fields in Hadamard manifolds, Set-Valued Anal, 19, 3, 361-383 (2011) · Zbl 1239.47043 · doi:10.1007/s11228-010-0169-1 |
| [20] | Ansari, Q. H.; Islam, M.; Yao, J. C., Nonsmooth variational inequalities on Hadamard manifolds, Appl. Anal. V, 99, 2, 340C358 (2020) · Zbl 1431.49003 |
| [21] | da Cruz Neto, J. X.; Ferreira, O. P.; Lucambio., Pérez, L. R., Monotone point-to-set vector fields, Balkan J. Geometry Appl, 5, 1, 69-79 (2000) · Zbl 0978.90076 |
| [22] | Colao, V.; López, G.; Marino, G.; Martín-Márquez, V., Equilibrium problems in Hadamard manifolds, J. Math. Anal. Appl, 388, 1, 61-71 (2012) · Zbl 1273.49015 · doi:10.1016/j.jmaa.2011.11.001 |
| [23] | Ceng, L. C.; Li, X.; Qin, X., Parallel proximal point methods for systems of vector optimization problems on Hadamard manifolds without convexity, Optimization., 69, 2, 357-383 (2020) · Zbl 1432.90120 · doi:10.1080/02331934.2019.1625354 |
| [24] | Shih-Sen Chang, Tang, J. F., Wen, C F. (2021). New algorithm for fixed points and monotone inclusion problems on Hadamard manifolds, Acta Mathematica Scientia, English Series (in print). |
| [25] | Wang, X.; López, G.; Li, C.; Yao, J. C., Equilibrium problems on Riemannian mani-folds with applications, J Math Anal Appl, 473, 2, 866-891 (2019) · Zbl 1414.53034 · doi:10.1016/j.jmaa.2018.12.073 |
| [26] | Yao, Y.; Postolache, M.; Liou, Y. C.; Yao, Z., Construction algorithms for a class of monotone variational inequalities, Optim. Lett., 10, 7, 1519-1528 (2016) · Zbl 1380.90266 · doi:10.1007/s11590-015-0954-8 |
| [27] | Sakai, T., Riemannian Geometry, Translations of Mathematical Monographs (1996), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0886.53002 |
| [28] | Ferreira, O. P.; Oliveira, P. R., Proximal point algorithm on Riemannian manifolds, Optimization, 51, 2, 257-270 (2002) · Zbl 1013.49024 · doi:10.1080/02331930290019413 |
| [29] | Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci, 2, 1, 183-202 (2009) · Zbl 1175.94009 · doi:10.1137/080716542 |
| [30] | Chang, S. S.; Wen, C. F.; Yao, J. C., Common zero point for a finite family of inclusion problems of accretive mappings in banach spaces, Optimization, 67, 8, 1183-1196 (2018) · Zbl 1402.90119 · doi:10.1080/02331934.2018.1470176 |
| [31] | Yunier, J.; Cruz, B., On proximal subgradient splitting method for minimizing the sum of two nonsmooth convex functions, Set-Val. Var. Anal, 25, 245-263 (2017) · Zbl 1365.65160 |
| [32] | Zhao, X.; Ng, K. F.; Li, C.; Yao, J. C., Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems, Appl. Math. Optim., 78, 3, 613-641 (2018) · Zbl 1492.47089 · doi:10.1007/s00245-017-9417-1 |
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