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doi: 10.3934/naco.2021040
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Geodesic $ \mathcal{E} $-prequasi-invex function and its applications to non-linear programming problems

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

*Corresponding author: Akhlad Iqbal

Received  March 2020 Revised  August 2021 Early access September 2021

In this article, we define a new class of functions on Riemannian manifolds, called geodesic $ \mathcal{E} $-prequasi-invex functions. By a suitable example it has been shown that it is more generalized class of convex functions. Some of its characteristics are studied on a nonlinear programming problem. We also define a new class of sets, named geodesic slack invex set. Furthermore, a sufficient optimality condition is obtained for a nonlinear programming problem defined on a geodesic local $ \mathcal{E} $-invex set.

Citation: Akhlad Iqbal, Praveen Kumar. Geodesic $ \mathcal{E} $-prequasi-invex function and its applications to non-linear programming problems. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021040
References:
[1]

I. Ahmad, A. Iqbal and Shahid Ali, On properties of geodesic $\eta$-preinvex functions, Advances in Operations Research, (2009), Article ID 381831, 10 pages. doi: 10.1155/2009/381831.

[2]

A. Barani and M. R. Pouryayevali, Invex sets and preinvex functions on Riemannian manifolds, J. Math. Anal. App., 328 (2007), 767-779.  doi: 10.1016/j.jmaa.2006.05.081.

[3]

D. I. DucaE. DucaL. Lupsa and R. Blaga, E-convexfunctions, Bull. Appl. Comput. Math., 43 (2000), 93-103. 

[4]

D. I. Duca and L. Lupsa, On the E-epigraph of an E-convex function, Journal of Optimization Theory and Applications, 129 (2006), 341-348.  doi: 10.1007/s10957-006-9059-y.

[5]

C. Fulga and V. Preda, Nonlinear programming with E-preinvex and local E-preinvex functions, Eur. J. Oper. Res., 192 (2009), 737-743.  doi: 10.1016/j.ejor.2007.11.056.

[6]

M. A. Hanson, On sufficiency of Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981) 545–550. doi: 10.1016/0022-247X(81)90123-2.

[7]

A. Iqbal and P. Kumar, On slack 2-geodesic convex sets and geodesic E-pseudoconvex functions, preprint.

[8]

A. IqbalS. 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: 10.1007/s10957-012-0052-3.

[9]

A. Kilicman and W. Saleh, On properties of geodesic semilocal $E$-preinvex functions, Journal of Inequalities and Applications, (2018), Article number: 353. doi: 10.1186/s13660-018-1944-z.

[10]

A. Kilicman and W. Saleh, Generalized preinvex functions and their applications, Symmetry, 10 (2018), 493.  doi: 10.3390/sym10100493.

[11]

B. Kumari and A. Jayswal, Some properties of geodesic $E$-preinvex function and geodesic semi $E$-preinvex function on Reimannian manifolds, Opsearch, 55 (2018), 807-822.  doi: 10.1007/s12597-018-0346-9.

[12]

S. Lang, Fundamentals of Differential Geometry, Grad. Texts in Math., Springer, New York, 1999. doi: 10.1007/978-1-4612-0541-8.

[13]

S. Mititelu, Generalized invexity and vector optimization on differential manifolds, Diff. Geom. Dynam. Syst., 3 (2001), 21-31. 

[14]

R. Pini, Convexity along curves and invexity, Optimization, 29 (1994), 301-330.  doi: 10.1080/02331939408843959.

[15]

T. Rapcsak, Smooth Nonlinear Optimization in $R^n$, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6357-0.

[16]

W. Saleh and A. Kilicman, On the characteristic properties of geodesic Sub-($\alpha$, b, s)-preinvex, Preprints 2019, 2019100301. doi: 10.20944/preprints201910.0301.v1.

[17]

Y. R. Syau and E. S. Lee, Some properties of E-convex functions, Appl. Math. Lett., 18 (2005), 1074-1080.  doi: 10.1016/j.aml.2004.09.018.

[18]

C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic, Amsterdam, 1994. doi: 10.1007/978-94-015-8390-9.

[19]

X. M. Yang, On E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl., 109 (2001), 699-704.  doi: 10.1023/A:1017532225395.

[20]

E. A. Youness, E-convex sets, E-convex functions, and E-convex programming, Journal of Optimization Theory and Applications, 102 (1999), 439-450.  doi: 10.1023/A:1021792726715.

show all references

References:
[1]

I. Ahmad, A. Iqbal and Shahid Ali, On properties of geodesic $\eta$-preinvex functions, Advances in Operations Research, (2009), Article ID 381831, 10 pages. doi: 10.1155/2009/381831.

[2]

A. Barani and M. R. Pouryayevali, Invex sets and preinvex functions on Riemannian manifolds, J. Math. Anal. App., 328 (2007), 767-779.  doi: 10.1016/j.jmaa.2006.05.081.

[3]

D. I. DucaE. DucaL. Lupsa and R. Blaga, E-convexfunctions, Bull. Appl. Comput. Math., 43 (2000), 93-103. 

[4]

D. I. Duca and L. Lupsa, On the E-epigraph of an E-convex function, Journal of Optimization Theory and Applications, 129 (2006), 341-348.  doi: 10.1007/s10957-006-9059-y.

[5]

C. Fulga and V. Preda, Nonlinear programming with E-preinvex and local E-preinvex functions, Eur. J. Oper. Res., 192 (2009), 737-743.  doi: 10.1016/j.ejor.2007.11.056.

[6]

M. A. Hanson, On sufficiency of Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981) 545–550. doi: 10.1016/0022-247X(81)90123-2.

[7]

A. Iqbal and P. Kumar, On slack 2-geodesic convex sets and geodesic E-pseudoconvex functions, preprint.

[8]

A. IqbalS. 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: 10.1007/s10957-012-0052-3.

[9]

A. Kilicman and W. Saleh, On properties of geodesic semilocal $E$-preinvex functions, Journal of Inequalities and Applications, (2018), Article number: 353. doi: 10.1186/s13660-018-1944-z.

[10]

A. Kilicman and W. Saleh, Generalized preinvex functions and their applications, Symmetry, 10 (2018), 493.  doi: 10.3390/sym10100493.

[11]

B. Kumari and A. Jayswal, Some properties of geodesic $E$-preinvex function and geodesic semi $E$-preinvex function on Reimannian manifolds, Opsearch, 55 (2018), 807-822.  doi: 10.1007/s12597-018-0346-9.

[12]

S. Lang, Fundamentals of Differential Geometry, Grad. Texts in Math., Springer, New York, 1999. doi: 10.1007/978-1-4612-0541-8.

[13]

S. Mititelu, Generalized invexity and vector optimization on differential manifolds, Diff. Geom. Dynam. Syst., 3 (2001), 21-31. 

[14]

R. Pini, Convexity along curves and invexity, Optimization, 29 (1994), 301-330.  doi: 10.1080/02331939408843959.

[15]

T. Rapcsak, Smooth Nonlinear Optimization in $R^n$, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6357-0.

[16]

W. Saleh and A. Kilicman, On the characteristic properties of geodesic Sub-($\alpha$, b, s)-preinvex, Preprints 2019, 2019100301. doi: 10.20944/preprints201910.0301.v1.

[17]

Y. R. Syau and E. S. Lee, Some properties of E-convex functions, Appl. Math. Lett., 18 (2005), 1074-1080.  doi: 10.1016/j.aml.2004.09.018.

[18]

C. Udriste, Convex Functions and Optimization Methods on Riemannian Manifolds, Kluwer Academic, Amsterdam, 1994. doi: 10.1007/978-94-015-8390-9.

[19]

X. M. Yang, On E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl., 109 (2001), 699-704.  doi: 10.1023/A:1017532225395.

[20]

E. A. Youness, E-convex sets, E-convex functions, and E-convex programming, Journal of Optimization Theory and Applications, 102 (1999), 439-450.  doi: 10.1023/A:1021792726715.

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