Abstract
In the present paper, the notion of geodesic E-preinvex function and geodesic semi E-preinvex function are introduced on Riemannian manifold. Moreover, several properties, results and relations are studied within aforesaid functions. An example is also constructed to illustrate the definition of geodesic E-preinvex function. In addition, we have studied the optimality results with the help of geodesic E-preinvex and geodesic semi E-preinvex functions.
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Agarwal, R.P., Ahmad, I., Iqbal, A., Ali, S.: Generalized invex sets and preinvex functions on Riemannian manifolds. Taiwan. J. Math. 16, 1719–1732 (2012)
Ahmad, I., Iqbal, A., Ali, S.: On properties of geodesic \(\eta \)-preinvex functions. Adv. Oper. Res. 2009, 1–10 (2009)
Antczak, T.: \(r\)-Preinvexity and \(r\)-invexity in mathematical programming. Comput. Math. Appl. 50, 551–566 (2005)
Barani, A., Pouryayevali, M.R.: Invex sets and preinvex functions on Riemannian manifolds. J. Math. Anal. Appl. 328, 767–779 (2007)
Ben-Israel, A., Mond, B.: What is the invexity. J. Aust. Math. Soc. 28, 1–9 (1986)
Duca, D.I., Duca, E., Lupsa, L., Blaga, R.: \(E\)-convex functions. Bull. Appl. Comput. Math. 43, 93–103 (2000)
Duca, D.I., Lupsa, L.: On the E-epigraph of an \(E\)-convex function. J. Optim. Theory Appl. 120, 341–346 (2006)
Fulga, C., Preda, V.: Nonlinear programming with \(E\)-preinvex and local \(E\)-preinvex functions. Eur. J. Oper. Res. 192, 737–743 (2009)
Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Iqbal, A., Ahmad, I., Ali, S.: Some properties of geodesic semi-\(E\)-convex functions. Nonlinear Anal. 74, 6805–6813 (2011)
Iqbal, A., Ali, S., Ahmad, I.: On geodesic \(E\)-convex sets, geodesic \(E\)-convex functions and E-epigraph. J. Optim. Theory Appl. 55, 239–251 (2012)
Jeyakumar, V.: Strong and weak invexity in mathematical programming. Math. Oper. Res. 55, 109–125 (1985)
Kiliçman, A., Saleh, W.: On geodesic strongly \(E\)-convex sets and geodesic strongly \(E\)-convex functions. J. Ineq. Appl. 2015, 1–10 (2015)
Kiliçman, A., Saleh, W.: Some properties of geodesic semi \(E\)-\(b\)-Vex function. Open Math. 13, 795–804 (2015)
Kiliçman, A., Saleh, W.: On geodesic semi strongly \(E\)-convex functions. J. Interdiscip. Math. 19, 1039–1055 (2016)
Lang, S.: Fundamental of Differential Geometry, Graduate Texts in Mathematics. Springer, New York (1999)
Mititelu, S.: Generalized invexity and vector optimization on differential manifolds. Differ. Geom. Dyn. Syst. 3, 21–31 (2001)
Noor, M.A.: Variational-like inequality. Optimization 30, 323–330 (1994)
Pini, R.: Convexity along curves and invexity. Optimization 29, 301–309 (1994)
Porwal, S.K.: Geodesic semilocal \(E\)-\(b\)-Vex functions on Riemannian manifolds. Adv. Model. Optim. 20, 395–405 (2018)
Rapcsak, T.: Smooth Nonlinear Optimization in \(R^n\). Kluwer Academic, Dordrecht (1997)
Syau, Y.R., Lee, E.S.: Some properties of \(E\)-convex functions. Appl. Math. Lett. 18, 1074–1080 (2005)
Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic, Dordrecht (1994). (Math. Appl.)
Weir, T., Mond, B.: Preinvex functions in multiobjective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)
Yang, X.M.: On \(E\)-convex sets, \(E\)-convex functions and \(E\)-convex programming. J. Optim. Theory Appl. 109, 699–704 (2001)
Youness, E.A.: On \(E\)-convex sets, \(E\)-convex functions and \(E\)-convex programming. J. Optim. Theory Appl. 102, 439–450 (1999)
Acknowledgements
The authors are thankful to referees for their valuable remarks which improved the results and presentation of this article. This work is financially supported by the Council of Scientific and Industrial Research, New Delhi, India through Grant No.: 25(0266)/17/EMR-II.
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Kumari, B., Jayswal, A. Some properties of geodesic E-preinvex function and geodesic semi E-preinvex function on Riemannian manifolds. OPSEARCH 55, 807–822 (2018). https://doi.org/10.1007/s12597-018-0346-9
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DOI: https://doi.org/10.1007/s12597-018-0346-9
Keywords
- Geodesic E-invex set
- Geodesic E-preinvex function
- Geodesic semi E-preinvex function
- Riemannian manifold