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On Geodesic E-Convex Sets, Geodesic E-Convex Functions and E-Epigraphs

Author

Listed:
  • Akhlad Iqbal

    (Aligarh Muslim University
    BITS Pilani Hyderabad Campus)

  • Shahid Ali

    (Aligarh Muslim University)

  • I. Ahmad

    (Aligarh Muslim University
    King Fahd University of Petroleum and Minerals)

Abstract

In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R n is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.

Suggested Citation

  • Akhlad Iqbal & Shahid Ali & I. Ahmad, 2012. "On Geodesic E-Convex Sets, Geodesic E-Convex Functions and E-Epigraphs," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 239-251, October.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:1:d:10.1007_s10957-012-0052-3
    DOI: 10.1007/s10957-012-0052-3
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    References listed on IDEAS

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    1. X. M. Yang, 2001. "On E-Convex Sets, E-Convex Functions, and E-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 699-704, June.
    2. D. I. Duca & L. Lupşa, 2006. "On the E-Epigraph of an E-Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 129(2), pages 341-348, May.
    3. X. M. Yang & X. Q. Yang & K. L. Teo, 2001. "Characterizations and Applications of Prequasi-Invex Functions," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 645-668, September.
    4. E. A. Youness, 1999. "E-Convex Sets, E-Convex Functions, and E-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 439-450, August.
    5. X.M. Yang & X.Q. Yang & K.L. Teo, 2003. "Generalized Invexity and Generalized Invariant Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 117(3), pages 607-625, June.
    6. Fulga, C. & Preda, V., 2009. "Nonlinear programming with E-preinvex and local E-preinvex functions," European Journal of Operational Research, Elsevier, vol. 192(3), pages 737-743, February.
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    Cited by:

    1. Babli Kumari & Anurag Jayswal, 2018. "Some properties of geodesic E-preinvex function and geodesic semi E-preinvex function on Riemannian manifolds," OPSEARCH, Springer;Operational Research Society of India, vol. 55(3), pages 807-822, November.
    2. O. Ferreira & A. Iusem & S. Németh, 2014. "Concepts and techniques of optimization on the sphere," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 1148-1170, October.
    3. Izhar Ahmad & Meraj Ali Khan & Amira A. Ishan, 2019. "Generalized Geodesic Convexity on Riemannian Manifolds," Mathematics, MDPI, vol. 7(6), pages 1-12, June.

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