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November  2020, 16(6): 2971-2989. doi: 10.3934/jimo.2019089

Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function

Tadeusz Antczak 1, and  Najeeb Abdulaleem 2,,

1. 

Faculty of Mathematics and Computer Science University of Łódź, Banacha 22, 90-238 Łódź, Poland
2. 

Department of Mathematics, Hadhramout University, P.O. BOX : (50511-50512), Al-Mahrah, Yemen

* Corresponding author: Najeeb Abdulaleem

Received  July 2018 Revised  March 2019 Published  July 2019

In this paper, a nonconvex vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. The functions constituting it are not necessarily differentiable, but they are $ E $-differentiable. The so-called $ E $-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered $ E $-differentiable vector optimization problem with the multiple interval-valued objective function. Also the sufficient optimality conditions are derived for such interval-valued vector optimization problems under appropriate (generalized) $ E $-convexity hypotheses.

Keywords: $ E $-differentiable function, $ E $-differentiable vector optimization problem with multiple interval-valued objective function, $ E $-Karush-Kuhn-Tucker necessary optimality conditions, $ E $-convex function.
Mathematics Subject Classification: Primary: 90C29, 90C30, 90C46, 90C26.
Citation: Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089
References:
[1]

I. Ahmad, A. Jayswal and J. Banerjee, On interval-valued optimization problems with generalized invex functions, J. of Inequalities and Applications, 313 (2013), 14pp. doi: 10.1186/1029-242X-2013-313.  Google Scholar

[2]

I. Ahmad, D. Singh and B. A. Dar, Optimality conditions for invex interval valued nonlinear programming problems involving generalized H-derivative, Filomat, 30 (2016), 2121-2138.  doi: 10.2298/FIL1608121A.  Google Scholar

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G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, NY, 1983.  Google Scholar

[4]

A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Math. Methods of Oper. Research, 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0.  Google Scholar

[5]

S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions – a generalized approach, European J. of Oper. Research, 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0.  Google Scholar

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X. Chen, Some properties of semi-E-convex functions, J. of Math. Anal. and Applications, 275 (2002), 251-262.  doi: 10.1016/S0022-247X(02)00325-6.  Google Scholar

[7]

X. Chen and Z. Li, On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity, J. of Indust. and Mgmt. Optimization, 14 (2018), 895-912.  doi: 10.3934/jimo.2017081.  Google Scholar

[8]

D. I. Duca, E. Duca, L. Lupsa and R. Blaga, E-convex functions, Bulletin of Appl. Computational Math., 43 (2000), 93-103.   Google Scholar

[9]

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

[10]

T. Emam and E. A. Youness, Semi strongly E-convex function, J. of Math. and Statistics, 1 (2005), 51-57.  doi: 10.3844/jmssp.2005.51.57.  Google Scholar

[11]

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

[12]

J. S. Grace and P. Thangavelu, Properties of E-convex sets, Tamsui Oxford J. of Math. Sciences, 25 (2009), 1-7.   Google Scholar

[13]

E. Hosseinzade and H. Hassanpour, The Karush-Kuhn-Tucker optimality conditions in interval-valued multiobjective programming problems, J. of Appl. Math. & Informatics, 29 (2011), 1157-1165.   Google Scholar

[14]

M. Jana and G. Panda, Solution of nonlinear interval vector optimization problem, Oper. Research, 14 (2014), 71-85.   Google Scholar

[15]

A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Appl. Math. and Computation, 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041.  Google Scholar

[16]

S. Karmakar and A. K. Bhunia, An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming, J. of the Egyptian Math. Society, 22 (2014), 292-303.  doi: 10.1016/j.joems.2013.07.002.  Google Scholar

[17]

L. Li, S. Liu and J. Zhang, Univex interval-valued mapping with differentiability and its application in nonlinear programming, J. of Appl. Math., 2013 (2013), 8pp. doi: 10.1155/2013/383692.  Google Scholar

[18]

L. Li, S. Liu and J. Zhang, On interval-valued invex mappings and optimality conditions for interval-valued optimization problems, J. of Ineq. and Appl., 2015 (2015), 19pp. doi: 10.1186/s13660-015-0692-6.  Google Scholar

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L. Lupsa and D. I. Duca, E-convex programming, Revue d'Analyse Numerique et de Theorie de l'Approximation, 33 (2004), 183-187.   Google Scholar

[20]

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.  Google Scholar

[21]

A. E.-M. A. Megahed, H. G. Gomma, E. A. Youness and A.-Z. H. El-Banna, Optimality conditions of E-convex programming for an E-differentiable function, J. of Ineq. and Appl., 2013 (2013), 11pp. doi: 10.1186/1029-242X-2013-246.  Google Scholar

[22]

F. Mirzapour, Some properties on E-convex and E-quasi-convex functions, in The 18th Seminar on Mathematical Analysis and its Applications, 26-27 Farvardin, 1388, Tarbiat Moallem University, 2009, 178–181. Google Scholar

[23]

R. E. Moore, Method and Applications of Interval Analysis, SIAM, Philadelphia, 1979.  Google Scholar

[24]

G.-R. Piao, L. Jiao and D. S. Kim, Optimality and mixed duality in multiobjective E-convex programming, J. of Ineq. and Appl., 2015 (2015), 13pp. doi: 10.1186/s13660-015-0854-6.  Google Scholar

[25]

D. Singh, B. A. Dar and A. Goyal, KKT optimality conditions for interval valued optimization problems, J. of Nonlinear Anal. and Optimization, 5 (2014), 91-103.   Google Scholar

[26]

D. Singh, B. A. Dar and D. S. Kim, KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions, European J. of Oper. Research, 254 (2016), 29-39.  doi: 10.1016/j.ejor.2016.03.042.  Google Scholar

[27]

M. Soleimani-Damaneh, E-convexity and its generalizations, Int. J. of Computer Math., 88 (2011), 3335-3349.  doi: 10.1080/00207160.2011.589899.  Google Scholar

[28]

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

[29]

H.-C. Wu, On interval-valued nonlinear programming problems, J. of Math. Anal. and Applications, 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[30]

H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. of Oper. Research, 196 (2009), 49-60.  doi: 10.1016/j.ejor.2008.03.012.  Google Scholar

[31]

X. M. Yang, On E-convex sets, E-convex functions, and E-convex programing, J. of Optimization Theory and Applications, 109 (2001), 699-704.  doi: 10.1023/A:1017532225395.  Google Scholar

[32]

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

[33]

E. A. Youness, Optimality criteria in E-convex programming, Chaos, Solitons & Fractals, 12 (2001), 1737-1745.  doi: 10.1016/S0960-0779(00)00036-9.  Google Scholar

[34]

E. A. Youness, Characterization of efficient solution of multiobjective E-convex programming problems, Appl. Math. and Computation, 151 (2004), 755-761.  doi: 10.1016/S0096-3003(03)00526-5.  Google Scholar

[35]

J. K. Zhang, S. Y. Liu, L. F. Li and Q. X. Feng, The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optimization Letters, 8 (2014), 607-631.  doi: 10.1007/s11590-012-0601-6.  Google Scholar

[36]

H.-C. Zhou and Y.-J. Wang, Optimality condition and mixed duality for interval-valued optimization, in Fuzzy Information and Engineering, Vol. 2, Advances in Intelligent and Soft Computing, 62, Springer, 2009, 1315–1323. doi: 10.1007/978-3-642-03664-4_140.  Google Scholar

show all references

References:
[1]

I. Ahmad, A. Jayswal and J. Banerjee, On interval-valued optimization problems with generalized invex functions, J. of Inequalities and Applications, 313 (2013), 14pp. doi: 10.1186/1029-242X-2013-313.  Google Scholar

[2]

I. Ahmad, D. Singh and B. A. Dar, Optimality conditions for invex interval valued nonlinear programming problems involving generalized H-derivative, Filomat, 30 (2016), 2121-2138.  doi: 10.2298/FIL1608121A.  Google Scholar

[3]

G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, NY, 1983.  Google Scholar

[4]

A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Math. Methods of Oper. Research, 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0.  Google Scholar

[5]

S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions – a generalized approach, European J. of Oper. Research, 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0.  Google Scholar

[6]

X. Chen, Some properties of semi-E-convex functions, J. of Math. Anal. and Applications, 275 (2002), 251-262.  doi: 10.1016/S0022-247X(02)00325-6.  Google Scholar

[7]

X. Chen and Z. Li, On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity, J. of Indust. and Mgmt. Optimization, 14 (2018), 895-912.  doi: 10.3934/jimo.2017081.  Google Scholar

[8]

D. I. Duca, E. Duca, L. Lupsa and R. Blaga, E-convex functions, Bulletin of Appl. Computational Math., 43 (2000), 93-103.   Google Scholar

[9]

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

[10]

T. Emam and E. A. Youness, Semi strongly E-convex function, J. of Math. and Statistics, 1 (2005), 51-57.  doi: 10.3844/jmssp.2005.51.57.  Google Scholar

[11]

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

[12]

J. S. Grace and P. Thangavelu, Properties of E-convex sets, Tamsui Oxford J. of Math. Sciences, 25 (2009), 1-7.   Google Scholar

[13]

E. Hosseinzade and H. Hassanpour, The Karush-Kuhn-Tucker optimality conditions in interval-valued multiobjective programming problems, J. of Appl. Math. & Informatics, 29 (2011), 1157-1165.   Google Scholar

[14]

M. Jana and G. Panda, Solution of nonlinear interval vector optimization problem, Oper. Research, 14 (2014), 71-85.   Google Scholar

[15]

A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Appl. Math. and Computation, 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041.  Google Scholar

[16]

S. Karmakar and A. K. Bhunia, An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming, J. of the Egyptian Math. Society, 22 (2014), 292-303.  doi: 10.1016/j.joems.2013.07.002.  Google Scholar

[17]

L. Li, S. Liu and J. Zhang, Univex interval-valued mapping with differentiability and its application in nonlinear programming, J. of Appl. Math., 2013 (2013), 8pp. doi: 10.1155/2013/383692.  Google Scholar

[18]

L. Li, S. Liu and J. Zhang, On interval-valued invex mappings and optimality conditions for interval-valued optimization problems, J. of Ineq. and Appl., 2015 (2015), 19pp. doi: 10.1186/s13660-015-0692-6.  Google Scholar

[19]

L. Lupsa and D. I. Duca, E-convex programming, Revue d'Analyse Numerique et de Theorie de l'Approximation, 33 (2004), 183-187.   Google Scholar

[20]

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.  Google Scholar

[21]

A. E.-M. A. Megahed, H. G. Gomma, E. A. Youness and A.-Z. H. El-Banna, Optimality conditions of E-convex programming for an E-differentiable function, J. of Ineq. and Appl., 2013 (2013), 11pp. doi: 10.1186/1029-242X-2013-246.  Google Scholar

[22]

F. Mirzapour, Some properties on E-convex and E-quasi-convex functions, in The 18th Seminar on Mathematical Analysis and its Applications, 26-27 Farvardin, 1388, Tarbiat Moallem University, 2009, 178–181. Google Scholar

[23]

R. E. Moore, Method and Applications of Interval Analysis, SIAM, Philadelphia, 1979.  Google Scholar

[24]

G.-R. Piao, L. Jiao and D. S. Kim, Optimality and mixed duality in multiobjective E-convex programming, J. of Ineq. and Appl., 2015 (2015), 13pp. doi: 10.1186/s13660-015-0854-6.  Google Scholar

[25]

D. Singh, B. A. Dar and A. Goyal, KKT optimality conditions for interval valued optimization problems, J. of Nonlinear Anal. and Optimization, 5 (2014), 91-103.   Google Scholar

[26]

D. Singh, B. A. Dar and D. S. Kim, KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions, European J. of Oper. Research, 254 (2016), 29-39.  doi: 10.1016/j.ejor.2016.03.042.  Google Scholar

[27]

M. Soleimani-Damaneh, E-convexity and its generalizations, Int. J. of Computer Math., 88 (2011), 3335-3349.  doi: 10.1080/00207160.2011.589899.  Google Scholar

[28]

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

[29]

H.-C. Wu, On interval-valued nonlinear programming problems, J. of Math. Anal. and Applications, 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[30]

H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. of Oper. Research, 196 (2009), 49-60.  doi: 10.1016/j.ejor.2008.03.012.  Google Scholar

[31]

X. M. Yang, On E-convex sets, E-convex functions, and E-convex programing, J. of Optimization Theory and Applications, 109 (2001), 699-704.  doi: 10.1023/A:1017532225395.  Google Scholar

[32]

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

[33]

E. A. Youness, Optimality criteria in E-convex programming, Chaos, Solitons & Fractals, 12 (2001), 1737-1745.  doi: 10.1016/S0960-0779(00)00036-9.  Google Scholar

[34]

E. A. Youness, Characterization of efficient solution of multiobjective E-convex programming problems, Appl. Math. and Computation, 151 (2004), 755-761.  doi: 10.1016/S0096-3003(03)00526-5.  Google Scholar

[35]

J. K. Zhang, S. Y. Liu, L. F. Li and Q. X. Feng, The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optimization Letters, 8 (2014), 607-631.  doi: 10.1007/s11590-012-0601-6.  Google Scholar

[36]

H.-C. Zhou and Y.-J. Wang, Optimality condition and mixed duality for interval-valued optimization, in Fuzzy Information and Engineering, Vol. 2, Advances in Intelligent and Soft Computing, 62, Springer, 2009, 1315–1323. doi: 10.1007/978-3-642-03664-4_140.  Google Scholar

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