Abstract
In this paper, we consider the inclusion problems for maximal monotone set-valued vector fields defined on Hadamard manifolds. We discuss the equivalence between nonemptiness of solution set of the inclusion problem and the coercivity condition. The boundedness of solution set of the inclusion problem is studied. An application of our results to optimization problems in Hadamard manifolds is also presented.
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References
Ansari, Q.H., Islam, M., Yao, J.C.: Nonsmooth convexity and monotonicity in terms of a bifunction on Riemannian manifolds. J. Nonlinear Convex Anal. 18(4), 743–762 (2017)
Ansari, Q.H., Islam, M., Yao, J.C.: Nonsmooth variational inequalities on Hadamard manifolds. Appl. Anal. (2018). https://doi.org/10.1080/00036811.2018.1495329
Ansari, Q.H., Babu, F., Li, X.-B.: Variational inclusion problems in Hadamard manifolds. J. Nonlinear Convex Anal. 19(2), 219–237 (2018)
Bianchi, M., Hadjisavvas, N., Schaible, S.: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 122(1), 1–17 (2004)
Brézis, H., Lions, P.L.: Produits infinis de resolvants. Israel J. Math. 29, 329–345 (1970)
Bruck, R.E.: A strongly convergent iterative solution of \(0 \in U(x)\) for a maximal monotone operator in Hilbert space. J. Math. Anal. Appl. 48, 114–126 (1974)
Bruck, R.E., Reich, S.: A general convergence principle in nonlinear functional analysis. Nonlinear Anal. 4, 939–950 (1980)
Chang, S.S.: Set-valued variational inclusions in Banach spaces. J. Math. Anal. Appl. 248, 438–454 (2000)
Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86, 433–438 (1999)
da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balk. J. Geom. Appl. 5(1), 69–79 (2000)
da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contribution to the study of monotone vector fields. Acta Math. Hung. 94(4), 307–320 (2002)
do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)
Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)
Fereira, O.P., Pérez, L.R.L., Németh, S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Global Optim. 31, 133–151 (2005)
Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)
Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)
Li, C., López, G., Martín-Márquez, V.: Resolvents of set-valued monotone vector fields in Hadamard manifolds. Set Valued Anal. 19, 361–383 (2011)
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)
Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695–5706 (2009)
Németh, S.Z.: Monotone vector fields. Publ. Math. (Debr.) 54, 437–449 (1999)
Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Sahu, D.R., Ansari, Q.H., Yao, J.C.: The prox-Tikhonov-like forward-backward method and applications. Taiwan. J. Math. 19, 481–503 (2015)
Sakai, T.: Riemannian Geometry, Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1996)
Takahashi, S., Takahashi, W., Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)
Tang, G.-J., Huang, N.-J.: Korpelevich’s method for variational inequality problems on Hadamard manifolds. J. Global Optim. 54, 493–509 (2012)
Tang, G.-J., Wang, X., Liu, H.-W.: Projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence. Optimization 64, 1081–1096 (2015)
Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)
Walter, R.: On the metric projections onto convex sets in Riemannian spaces. Arch. Math. XXV, 91–98 (1974)
Wang, J., Li, C., López, G., Yao, J.C.: Convergence analysis of inexact proximal point algorithms on Hadamard manifolds. J. Global Optim. 61, 553–573 (2015)
Wang, J., Li, C., López, G., Yao, J.C.: Proximal point algorithm on Hadamard manifolds: linear convergence and finite termination. SIAM J. Optim. 26(4), 2696–2729 (2016)
Zhang, Y., He, Y., Jiang, Y.: Existence and boundedness of solutions to maximal monotone inclusion problem. Optim. Lett. 11, 1565–1570 (2017)
Acknowledgements
Authors are grateful to the references for their valuable suggestions and corrections. In this research, first author was supported by a research grant of DSR-SERB No. EMR/2016/005124.
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Ansari, Q.H., Babu, F. Existence and boundedness of solutions to inclusion problems for maximal monotone vector fields in Hadamard manifolds. Optim Lett 14, 711–727 (2020). https://doi.org/10.1007/s11590-018-01381-x
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DOI: https://doi.org/10.1007/s11590-018-01381-x
Keywords
- Inclusion problems
- Maximal monotone vector fields
- Coercivity conditions
- Existence results
- Boundedness of solution set
- Hadamard manifolds