doi: 10.3934/naco.2021040
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[1]

Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2020, 16 (4) : 2045-2063. doi: 10.3934/jimo.2019042

[2]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[3]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035

[4]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[5]

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063

[6]

Peter Benner, Ryan Lowe, Matthias Voigt. $\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 119-133. doi: 10.3934/naco.2018007

[7]

Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368

[8]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, 2021, 29 (3) : 2445-2456. doi: 10.3934/era.2020123

[9]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317

[10]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

[11]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027

[12]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079

[13]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

[14]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021

[15]

Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3603-3631. doi: 10.3934/jimo.2020135

[16]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129

[17]

Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030

[18]

Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $ S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007

[19]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278

[20]

Gang Wang, Yuan Zhang. $ Z $-eigenvalue exclusion theorems for tensors. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1987-1998. doi: 10.3934/jimo.2019039

 Impact Factor: 

Article outline

[Back to Top]