Taiwanese Journal of Mathematics


Li-wen Zhou and Nan-jing Huang

Full-text: Open access


In this paper, a new class of roughly geodesic $B$-invex sets, quasi roughly geodesic $B$-invex functions and pseudo roughly geodesic $B$-invex functions are introduced and studied on Hadamard manifolds by relaxing the definitions of geodesic convex sets and functions. Some properties of quasi roughly geodesic $B$-invex functions and pseudo roughly geodesic $B$-invex functions are proved on Hadamard manifolds. As applications, some sufficient and necessary conditions for optimal solution of the nonlinear programming problems involving the quasi roughly geodesic $B$-invex functions and the pseudo roughly geodesic $B$-invex functions are given on Hadamard manifolds. The Mond-weir type dual problems for the nonlinear programming problems are also considered on Hadamard manifolds.

Article information

Taiwanese J. Math., Volume 17, Number 3 (2013), 833-855.

First available in Project Euclid: 10 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

Hadamard manifold roughly geodesic $B$-invex set roughly geodesic $B$-invex function nonlinear optimization problem Mond-weir type dual


Zhou, Li-wen; Huang, Nan-jing. ROUGHLY GEODESIC $B$-INVEX AND OPTIMIZATION PROBLEM ON HADAMARD MANIFOLDS. Taiwanese J. Math. 17 (2013), no. 3, 833--855. doi:10.11650/tjm.17.2013.1937. https://projecteuclid.org/euclid.twjm/1499705986

Export citation


  • R. P. Agarwal and I. Ahmad, Akhlad Iqbal and Shahid Ali, Geodesic $G$-invex sets and semistrictly geodesic $\eta$-preinvex functions, Optimization, 61 (2012), 1169-1174.
  • I. Ahmad, Akhlad Iqbal and Shahid Ali, On properties of geodesic $\eta$-preinvex functions, Adv. in Oper. Res., 2009 (2009), DOI:10.1155/2009/381831.
  • D. Azagra, J. Ferrera and F. Lopez-Mesas, Nonsmooth analysis and Hamilton-Jaccobi equations on Riemannian manifolds, J. Func. Anal., 220 (2005), 304-361.
  • A. Barani and M. R. Pouryayevali, Invex sets and preinvex functions on Riemannian manifolds, J. Math. Anal. Appl., 328 (2007), 767-779.
  • R. Bartolo, A. Germinario and M. Sanchez, Convextiy of domains of Riemannian manifolds, Annals of Global Analysis and Geometry, 21 (2002), 63-84.
  • M. S. Bazaraa and C. M. Shetty, Nonlinear Programming-Theory and Algorithms, Wiley, New York, 1979.
  • C. R. Bector and C. Singh, $B$-vex functions, J. Optim. Theory Appl., 71 (1991), 237-253
  • C. R. Bector, S. K. Suneja and C. Lalitha, Generalized $B$-vex functions and generalized $B$-vex programming, J. Optim. Theory Appl., 76 (1993), 561-576.
  • A. A. Borisenko, E. Gallego and A. Reventos, Relation between area and volume for $\lambda$-convex sets in Hadamard manifolds. Diff. Geo. Appl., 14 (2001), 267-280.
  • I. Chavel, Riemannian Geometry-A Modern Introduction, Cambridge University Press, 1993.
  • T. Emam, Roughly $B$-invex programming problems, Calcolo, 48 (2011), 173-188.
  • O. P. Ferreira, Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds, Nonlinear Anal., 68 (2008), 1517-1528.
  • O. P. Ferreira, L. R. Lucambio Perez and S. Z. Nemeth, Singularities of monotone vector fields and an extragradient-type algorithm, J. Global Optim., 31 (2005), 133-151.
  • O. P. Ferreira and P. R. Oliveira, Proximal point algorithm on Riemannian manifolds, Optimization, 51 (2002), 257-270.
  • M. A. Hanson, On sufficiency of Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550
  • C. D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl., 156 (1991), 341-357.
  • R. N. Kaul and S. Kuar, Optimality criteria in nonlinear programming involving non convex functions, J. Math. Anal. Appl., 105 (1985), 104-112.
  • W. Klingenberg, A Course in Differential Geometry, Springer-Verlag, 1978.
  • Y. S. Ledyaev and Q. J. Zhu, Nonsmooth Analysis on Smooth Manifolds, Transactions American Mathematical Society, 2007.
  • S. L. Li, C. Li, Y. C. Liou and J. C. Yao, Existence of solutions for variational inequalities on Riemannian manifolds, Nonlinear Anal., 71 (2009), 5695-5706.
  • C. Li, B. S. Mordukhovich, J. H. Wang and J. C. Yao, Weak sharp minima on Riemannian manifolds, SIAM J. Optim., 21 (2011), 1523-1560.
  • C. Li, G. Lopez and V. Martin-Marquez, Iterative algorithms for nonexpansive mappings on Hadamard manifolds, Taiwanese J. Math., 14 (2010), 541-559.
  • C. Li, G. Lopez and V. Martin-Marquez, Monotone vector fields and the proximal point algorithm on Hadamard manifolds, J. London Math. Soc., 79(2) (2009), 663-683.
  • S. Mititelu, Generalized invexity and vector optimization on differentiable manifolds, Differential Geometry-Dynamical Systems, 3 (2001), 21-31.
  • T. Morsy, A Study on Generalized Convex Mathematical Programming Problems, Master Thesis, Faculty of Science, Suez Canal University, Egypt, 2003.
  • S. Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal., 52 (2003), 1491-1498.
  • H. X. Phu, Six kinds of roughly convex functions, J. Optim. Theory Appl., 92 (1997), 357-375.
  • R. Pini, Convextiy along curves and invexity, Optimization, 29 (1994), 301-309.
  • T. Rapcsak, Smooth Nonlinear Optimization in $R^n$, Kluwer Academic Publishers, 1997.
  • T. Rapcsak, Geodesic convexity in nonlinear optimization, J. Optim. Theory Appl., 69 (1991), 169-183.
  • S. K. Sunela, C. Singh and C. R. Bector, Generalization of preinvex and $B$-vex functions, J. Optim. Theory Appl., 76(3) (1993), 277-287
  • G. J. Tang, L. W. Zhou and N. J. Huang, The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds, Optim. Lett., (2012), DOI: 10.1007/s11590-012-0459-7.
  • C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, in: Mathematics and its Applications, Vol. 297, Kluwer Academic Publishers, 1994.
  • C. Udriste, Kuhn-Tucker theorem on Riemannian manifolds, Colloquia Math. Soc. Janos Bolyai, Topics in Diff. Geometry, Debrecen, Hungary, 1984, pp. 1247-1259.
  • J. H. Wang, J. C. Yao and C. Li, Gauss-Newton methods for convex composite optimization on Riemannian manifolds, J. Glob. Optim., 53 (2012), 5-28.
  • L. W. Zhou and N. J. Huang, Existence of solutions for vector optimizations on Hadamard manifolds, J. Optim. Theory Appl., (2012), DOI: 10.1007/s10957-012-0186-3.