# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019089

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

 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.

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, 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 [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

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
 [1] Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010 [2] Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial & Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081 [3] Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 643-653. doi: 10.3934/naco.2013.3.643 [4] Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031 [5] Francesco Mainardi. On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2267-2278. doi: 10.3934/dcdsb.2014.19.2267 [6] Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615 [7] Alfonso Castro, Guillermo Reyes. Existence of multiple solutions for a semilinear problem and a counterexample by E. N. Dancer. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1135-1146. doi: 10.3934/cpaa.2017055 [8] Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131 [9] Baoxiang Wang. E-Besov spaces and dissipative equations. Communications on Pure & Applied Analysis, 2004, 3 (4) : 883-919. doi: 10.3934/cpaa.2004.3.883 [10] Tao Chen, Yunping Jiang, Gaofei Zhang. No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883 [11] Zutong Wang, Jiansheng Guo, Mingfa Zheng, Youshe Yang. A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle. Journal of Industrial & Management Optimization, 2015, 11 (1) : 13-26. doi: 10.3934/jimo.2015.11.13 [12] Florian Schneider. Second-order mixed-moment model with differentiable ansatz function in slab geometry. Kinetic & Related Models, 2018, 11 (5) : 1255-1276. doi: 10.3934/krm.2018049 [13] Augusto Visintin. P.D.E.s with hysteresis 30 years later. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 793-816. doi: 10.3934/dcdss.2015.8.793 [14] Alejandro Cataldo, Juan-Carlos Ferrer, Pablo A. Rey, Antoine Sauré. Design of a single window system for e-government services: the chilean case. Journal of Industrial & Management Optimization, 2018, 14 (2) : 561-582. doi: 10.3934/jimo.2017060 [15] M. Guru Prem Prasad, Tarakanta Nayak. Dynamics of { $\lambda tanh(e^z): \lambda \in R$\ ${ 0 }$ }. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 121-138. doi: 10.3934/dcds.2007.19.121 [16] Vladimir V. Marchenko, Klavdii V. Maslov, Dmitry Shepelsky, V. V. Zhikov. E.Ya.Khruslov. On the occasion of his 70th birthday. Networks & Heterogeneous Media, 2008, 3 (3) : 647-650. doi: 10.3934/nhm.2008.3.647 [17] Joan-Josep Climent, Juan Antonio López-Ramos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861-870. doi: 10.3934/amc.2016046 [18] Fei Gao. Data encryption algorithm for e-commerce platform based on blockchain technology. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1457-1470. doi: 10.3934/dcdss.2019100 [19] Caili Sang, Zhen Chen. $E$-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019042 [20] Zhanqiang Huo, Wuyi Yue, Naishuo Tian, Shunfu Jin. Performance evaluation for the sleep mode in the IEEE 802.16e based on a queueing model with close-down time and multiple vacations. Journal of Industrial & Management Optimization, 2009, 5 (3) : 511-524. doi: 10.3934/jimo.2009.5.511

2018 Impact Factor: 1.025

Article outline

[Back to Top]