Abstract
In this paper, we point out that a recent characterization of geodesic convex hull on Hadamard manifolds is not rigorous and explain why the characterization does not hold like it in linear spaces. Therefore, a definition of geodesic pseudo-convex combination is proposed to show that the Knaster–Kuratowski–Mazurkiewicz theorem still holds under some mild conditions on Hadamard manifolds.
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References
Rapcsák, T.: Smooth Nonlinear Optimization in \({\mathbb{R}}^n\). Kluwer, Dordrecht (1997)
Udriste, C.: Convex functions and optimization methods on Riemannian manifolds. In: Hazewinkel, M. (ed.) Mathematics and its Applications, vol. 297. Kluwer, Dordrecht (1994)
Rapcsák, T.: Geodesic convexity in nonlinear optimization. J. Optim. Theory Appl. 69, 169–183 (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)
Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Birkhauser Verlag, Basel (1997)
do Carmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)
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). Accessed 19 June 2009
Yang, Z., Pu, Y.J.: Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications. Nonlinear Anal. TMA 75, 516–525 (2012)
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)
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)
Kristály, A., Li, C., López, G., Nicolae, A.: What do ’convexities’ imply on Hadamard manifolds? J. Optim. Theory Appl. 170, 1068–1074 (2016)
Zhou, L.W., Xiao, Y.B., Huang, N.J.: New characterization of geodesic convexity on Hadamard manifolds with applications. J. Optim. Theory Appl. 172, 824–844 (2017)
Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. TMA 52, 1491–1498 (2003)
Tang, G.J., Zhou, L.W., Huang, N.J.: Existence results for a class of hemivariational inequality problems on Hadamard manifolds. Optimization 65, 1451–1461 (2016)
Chen, S.L., Huang, N.J.: Vector variational inequalities and vector optimization problems on Hadamard manifolds. Optim. Lett. 10, 753–767 (2016)
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)
Chavel, I.: Riemannian Geometry—A Modern Introduction. Cambridge University Press, London (1993)
Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)
Edwards, R.E.: Functional Analysis. Theory and Applications. Corrected reprint of the 1965 original. Dover Publications, Inc., New York (1995)
Németh, S.Z.: Variational inequalities on Hadamard manifolds. https://www.researchgate.net/publication/242990657/VariationalinequalitiesonHadamardmanifolds (2003). Accessed 27 Oct 2017
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)
Acknowledgements
The authors are grateful to the editor and the reviewers whose valuable comments and suggestions have led to much improvement of the paper. The authors would like to thank Professor Németh and Professor Kristály for their warm help. This work was supported by the National Natural Science Foundation of China (11471230, 11671282), the Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No.18TD0013) and the Youth Science and Technology Innovation Team of SWPU for Nonlinear Systems (No. 2017CXTD02).
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Communicated by Alexandru Kristály.
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Zhou, Lw., Huang, Nj. A Revision on Geodesic Pseudo-Convex Combination and Knaster–Kuratowski–Mazurkiewicz Theorem on Hadamard Manifolds. J Optim Theory Appl 182, 1186–1198 (2019). https://doi.org/10.1007/s10957-019-01511-0
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DOI: https://doi.org/10.1007/s10957-019-01511-0