A Projection-Type Method for Set Valued Variational Inequality Problems on Hadamard Manifolds
Tóm tắt
In this paper, we develop a projection-type algorithm for set-valued variational inequalities on Hadamard manifolds. The proposed method is well defined whether the solution set of the problem is non-empty or not. Under pseudomonotonicity assumptions on the underlying vector field, our method is convergent to a solution of the given set-valued variational inequality. The results presented in this paper generalize and improve some known results introduced by Tang et al. (Optimization 64(5):1081–1096, 2015).
Tài liệu tham khảo
Bai M.R., Zhou S.Z., Ni G.Y.: Variational-like inequalities with relaxed \({\eta -\alpha }\) pseudomonotone mappings in Banach spaces. Appl. Math. Lett. 19, 547–554 (2006)
Bai M.R., Zhou S.Z., Ni G.Y.: On the generalized monotonicity of variational inequalities. Comput. Math. Appl. 53, 910–917 (2007)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)
Colao V., Lopez G., Marino G., Martin-Marquez V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. 388, 61–77 (2012)
Crux Neto J.X., Ferreira O.P., Lucambio Perez L.R., Nemeth S.Z.: Convex- and monotone-transformable mathematical programming problems and a proximal-like point method. J. Glob. Optim. 35, 53–69 (2006)
Fang C.J., Chen S.L., Yang C.D.: An algorithm for solving multi-valued variational inequality. J. Inequal. Appl. 2013, 218 (2013)
Fang C.J., He Y.R.: A double projection algorithm for multi-valued variational inequalities and a unified framework of the method. Appl. Math. Comput. 217, 9543–9551 (2011)
Fang C.J., He Y.R.: An extragradient method for generalized variational inequality. Pac. J. Optim. 9(1), 47–59 (2013)
Fang C.J., Chen S.L.: A projection algorithm for set-valued variational inequalities on Hadamard manifolds. Optim. Lett. 9, 779–794 (2015)
Fang Y.P., Huang N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118, 327–338 (2003)
Farajzadeh A.P., Amini-Harandi A., Regan, D.O.: Existence results for generalized vector equilibrium problems with multivalued mappings via KKM theory. Abstr. Appl. Anal. (2008). doi:10.1155/2008/968478
Ferreira O.P., Pérez L.R.L., Németh S.Z.: Singularities of monotone vector fields and an extragradient-type algorithm. J. Glob. Optim. 31, 133–151 (2005)
Kinderlehrer D., Stampacchia G.: An introduction to variational inequalities and their applications. Academic Press, New York (1980)
Konnov, I.V.: Combined relaxation methods for varitional inequalities. Springer, Berlin (2001)
Li C., López G., Martín-Márquez V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79(2), 663–683 (2009)
Li, C., López, G., Martín-Márquez, V., Wang, J.H.: Resolvents of set-valued monotone vector fields in Hadamard manifolds. Set Valued Var. Anal. (2010). doi:10.1007/s11228-010-0169-1
Li S.L., Li C., Liou Y.C., Yao J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71(11), 5695–5706 (2009)
Li C., Yao J.C.: Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and thr proximal point algorithm. SIAM J. Control Optim. 50(4), 2486–2514 (2012)
Li, X.B., Huang, N.J.: Generalized vector quasi-equilibrium problems on Hadamard manifolds. Optim. Lett. 9, 155–170 (2015). doi:10.1007/s11590-013-0703-9
Luc D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 309–320 (2001)
Németh S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52(5), 1491–1498 (2003)
Németh S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)
Rapcsák T.: Nonconvex optimization and its applications, smooth nonlinear optimization in \({\mathbb{R}^{n}}\). Kluwer Academic Publishers, Dordrecht (1997)
Rapcsák T.: Geodesic convexity in nonlinear optimization. J. Optim. Theory Appl. 69, 169–183 (1991)
Sakai T.: Riemannian geometry, Translations of mathematical monographs, vol. 149. American Mathematical Society, Providence (1996)
Tang G.J., Huang N.J.: Korpelevich’s method for variational inequality problems on Hadamard manifolds. J. Glob. Optim. 54, 493–509 (2012)
Tang, G.J., Zhou, L.W., Huang, N.J.: The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds. Optim. Lett. 7, 779–790 (2013). doi:10.1007/s11590-012-0459-7
Tang G.J., Wang X., Liu H.W.: A projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence. Optimization 64(5), 1081–1096 (2015)
Udrişte, C.: Convex functions and optimization methods on riemannian manifolds. In: Mathematics and its Applications, vol. 297. Springer, Netherlands (1994)
Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146(3), 691708 (2010)
Willmore, T.J.: An introduction to differential geometry, Oxford University Press (1959)
Zhou, L.W., Huang, N.J.: Generalized KKM theorems on Hadamard manifolds with applications. (2009). http://www.paper.edu.cn/index.php/default/releasepaper/content/200906-669
Zhou L.W., Huang N.J.: Existence of solutions for vector optimization on Hadamard manifolds. J. Optim. Theory Appl. 157, 44–53 (2013)