Abstract
In this paper, some new results, concerned with the geodesic convex hull and geodesic convex combination, are given on Hadamard manifolds. An S-KKM theorem on a Hadamard manifold is also given in order to generalize the KKM theorem. As applications, a Fan–Browder-type fixed point theorem and a fixed point theorem for the a new mapping class are proved on Hadamard manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Klingenberg, W.: A Course in Differential Geometry. Springer-Verlag, Berlin (1978)
Rapcsák, T.: Smooth Nonlinear Optimization in \(\mathbb{R}^n\). Kluwer Academic Publishers, Dordrecht (1997)
Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and its Applications, vol. 297. Kluwer Academic Publishers, Dordrecht (1994)
Bazaraa, M.S., Shetty, C.M.: Nonlinear Programming-Theory and Algorithms. Wiley, New York (1979)
Rapcsák, T.: Geodesic convexity in nonlinear optimization. J. Optim. Theory Appl. 69, 169–183 (1991)
Horvath, C.D.: Contractibility and generalized convexity. J. Math. Anal. Appl. 156, 341–357 (1991)
Mititelu, S.: Generalized invexity and vector optimization on differentiable manifolds. Differ. Geom. Dyn. Syst. 3, 21–31 (2001)
Barani, A., Pouryayevali, M.R.: Invex sets and preinvex functions on Riemannian manifolds. J. Math. Anal. Appl. 328, 767–779 (2007)
Ferreira, O.P.: Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds. Nonlinear Anal. TMA. 68, 1517–1528 (2008)
Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007)
Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. TMA. 52, 1491–1498 (2003)
Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. TMA. 71, 5695–5706 (2009)
Colao, V., López, G., Marino, G., Martín-Márquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. 388, 61–77 (2012)
Zhou, L.W., Huang, N.J.: Existence of solutions for vector optimizations on Hadamard manifolds. J. Optim. Theory Appl. 157, 44–53 (2012)
Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)
Ferreira, 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)
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.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math. 14, 541–559 (2010)
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)
Zhou, L.W., Huang, N.J.: Generalized KKM theorems on Hadamard manifolds with applications. http://www.paper.edu.cn/index.php/default/releasepaper/content/200906-669 (2009)
Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16, 49–69 (2009)
Yang, Z., Pu, Y.J.: Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications. Nonlinear Anal. 75, 516–525 (2012)
Kristály, A., Li, C., López, G., Nicolae, A.: What do ’convexities’ imply on Hadamard manifolds? J. Optim. Theory Appl. 170, 1068–1074 (2016)
Chang, T.H., Yen, C.L.: Generalized KKM theorem and its applications. Banyan Math. J. 3, 137–148 (1996)
Chang, T.H., Huang, Y.Y., Jeng, J.C., Kuo, K.H.: On \(S\)-KKM property and related topics. J. Math. Anal. Appl. 229, 212–227 (1999)
Lin, L.J., Chang, T.H.: \(S\)-KKM theorems, saddle points and minimax inequalities. Nonlinear Anal. TMA 34, 73–86 (1998)
Chavel, I.: Riemannian Geometry—A Modern Introduction. Cambridge University Press, London (1993)
Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Birkhauser Verlag, Basel (1997)
do Carmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)
Sakai, T.: Riemannian Geometry. Transl. Math. Monogr., vol. 149. American Mathematical Society, Providence (1996)
Edward, R.E.: Functional Analysis: Theory and Applications. Dover Publications, New York (1995)
Acknowledgments
The authors are grateful to the editor and the referees for their valuable comments and suggestions. The authors also thank to Professor S.Z. Németh and Professor Chang T. H. for their warm help. This work was supported by the National Natural Science Foundation of China (11101069, 11471230, 11671282, 61310306022), Special Foundation of Sichuan Provincial Education Department (020402000044), the China Postdoctoral Science Foundation (2015T80967), the Applied Basic Project of Sichuan Province (2016JY0170), the Postdoctoral Science Foundation of Sichuan Province, the Key Program of Education Department of Sichuan Province, and the Fundamental Research Funds for the Central Universities (ZYGX2015J098).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Juan-Enrique Martinez Legaz.
Rights and permissions
About this article
Cite this article
Zhou, Lw., Xiao, Yb. & Huang, Nj. New Characterization of Geodesic Convexity on Hadamard Manifolds with Applications. J Optim Theory Appl 172, 824–844 (2017). https://doi.org/10.1007/s10957-016-1012-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-1012-0