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http://dx.doi.org/10.14317/jami.2018.521

GEODESIC SEMI E-PREINVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS  

PORWAL, SANDEEP KUMAR (department of Mathematics in Future Institute of Engineering & Technology)
Publication Information
Journal of applied mathematics & informatics / v.36, no.5_6, 2018 , pp. 521-530 More about this Journal
Abstract
Several classes of functions, named as semi E-preinvex functions and semilocal E-preinvex functions and their properties are studied by various authors. In this paper we introduce the geodesic concept over two types of problems first is semi E-preinvex functions and another is semilocal E-preinvex functions on Riemannian manifolds and study some of their properties.
Keywords
Geodesic E-invex set; Geodesic E-preinvex functions; Geodesic semi E-preinvex functions; Geodesic G-E-invex set; Geodesic local E-preinvex functions; Geodesic semilocal E-preinvex functions;
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1 T. Weir and B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl. 136 (1988), 29-38.   DOI
2 X. Yang and D. Li, Semistrictly preinvex functions, J. Math. Anal. Appl. 258 (2001), 287-308.   DOI
3 B.D. Craven, Duality for Generalized Convex Fractional Programs, in: Generalized concavity in optimization and Economic, Academic Press, New York, 1981.
4 D.I. Duca, E. Duca, L. Lupsa and R. Blaga, E-convex functions, Bull. Appl. Comput. Math. 43 (2000), 93-102.
5 D.I. Duca and L. Lupsa, On the E-epigraph of an E-convex function, J. Optim. Theory Appl. 120 (2006), 341-348.
6 G.M. Ewing, Sufficient conditions for global minima of suitably convex functionals from variational and control theory, SIAM Review 19, no. 2(1977), 202-220.   DOI
7 C. Fulga and V. Preda, Nonlinear programming with E-preinvex and local E-preinvex functions, Eur J Oper Res. 192 (2009), 737-743.   DOI
8 M.A. Hanson, On sufficiency of the KuhnTucker conditions, J. Math. Anal. Appl. 80 (1981), 545-550.   DOI
9 A. Iqbal, S. Ali and I. Ahmad, On geodesic E-convex sets, geodesic E-convex functions and E-epigraphs, J. Optim Theory Appl. 155 (2012), 239-251.   DOI
10 T. Weir and V. Jeyakumar, A class of nonconvex functions and mathematical programming, Bull. Aust. Math. Soc. 38 (1988), 177-189.   DOI
11 A. Iqbal, I. Ahmad and S. Ali, Some properties of geodesic semi E- convex functions, Nonlinear Analysis 74 (2011), 6805-6813.   DOI
12 Hu, QJ, Jian, JB, Zheng, HY, Tang, CM, Semilocal E-convexity and semilocal E-convex programming, Bull Aust Math Soc. 75 (2007), 59-74 .   DOI
13 H. Jiao, A class of semilocal E-preinvex maps in Banach spaces with applications to nondifferentiable vector optimization, IJOCTA 4 (2014), 1-10.
14 A. Ben-Israel and B. Mond, What is invexity?, J. Aust. Math. Soci., Series B 28 (1986), 1-9.   DOI
15 X.S. Chen, Some properties of semi E-convex functions, J Math Anal Appl. 275 (2002), 251-262.   DOI
16 R.N. Kaul and S. Kaur, Generalizations of convex and related functions, European Journal of Operational Research 9, no. 4(1982), 369-377 .   DOI
17 R.N. Kaul and S. Kaur, Sufficient optimality conditions using generalized convex functions, Opsearch 19, no. 4(1982), 212-224 .
18 S. Kaur, Theoretical studies in mathematical programming, Ph.D. Thesis, University of Delhi, New Delhi, India, 1983.
19 A. Barani and M.R. Pouryayevali, Invex sets and preinvex functions on Riemannian manifolds, J. Math. Anal. Appl. 328 (2007), 767-779.   DOI
20 A. Barani and M.R. Pouryayevali, Invariant monotone vector fields on Riemannian manifolds, Nonlinear Anal.: Theory Methods Appl. 70 (2009), 1850-1861.   DOI
21 C. Li, B.S. Mordukhovich, J. Wang, and J.C. Yao, Weak sharp minima of Riemannian manifolds, SIAM J. Optim. 21 (2011), 1523-1560.   DOI
22 Z.M. Luo and J.B. Jian, Some properties of semi E-preinvex maps in Banach spaces, Nonlinear Anal. Real World Appl. 12 (2011), 1243-1249 .   DOI
23 O.L. Mangasarian, Nonlinear Programming, McGraw Hill Book Company, New York, 1969.
24 S. Mititelu, Generalized invexity and vector optimization on differentiable manifolds, Differ. Geom. Dyn. Syst. 3 (2001), 21-31.
25 R. Pini, Convexity along curves and invexity, Optimization 29 (1994), 301-309.   DOI
26 V. Preda, I.M. Stancu-Minasian, and A. Batatorescu, Optimality and duality in nonlinear programming involving semilocally preinvex and related functions, Journal of Information & Optimization Sciences 17, no. 3(1996), 585-596 .   DOI
27 V. Preda and I.M. Stancu-Minasian, Duality in multiple objective programming involving semilocally preinvex and related functions, Glasnik Matematicki. Serija III, 32, no. 1(1997), 153-165 .
28 T. Rapcsak, Smooth Nonlinear Optimization in $\mathbb{R}^n$, Kluwer Academic Publishers, Dordrecht, 1997.
29 Yu-Ru Syau and E.S. Lee, Some properties of E-convex functions, Appl. Math. Lett. 18 (2005), 1074-1080.   DOI
30 C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic Publishers, Dordrecht, Netherlands, 1994.
31 W. Klingenberg, Riemannian geometry, Walter de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, 1982.
32 S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York, 1999.
33 X.M Yang, On E-convex sets, E-convex functions and E-convex programming, J. Optim. Theory Appl. 109 (2001), 699-704.   DOI
34 E.A. Youness, E-convex sets, On E-convex functions and E-convex programming, J. Optim Theory Appl. 102 (1999), 439-450.   DOI