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doi: 10.3934/naco.2021014

$V$-$E$-invexity in $E$-differentiable multiobjective programming

 Department of Mathematics, Hadhramout University, P.O. BOX : (50511-50512), Al-Mahrah, Yemen, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Received  June 2020 Revised  January 2021 Published  April 2021

In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems with $E$-differentiable functions. Namely, for an $E$-differentiable vector-valued function, the concept of $V$-$E$-invexity is defined as a generalization of the $E$-differentiable $E$-invexity notion and the concept of $V$-invexity. Further, the sufficiency of the so-called $E$-Karush-Kuhn-Tucker optimality conditions are established for the considered $E$-differentiable vector optimization problems with both inequality and equality constraints under $V$-$E$-invexity hypotheses. Furthermore, the so-called vector $E$-dual problem in the sense of Mond-Weir is defined for the considered $E$-differentiable multiobjective programming problem and several $E$-duality theorems are derived also under appropriate $V$-$E$-invexity assumptions.

Citation: Najeeb Abdulaleem. $V$-$E$-invexity in $E$-differentiable multiobjective programming. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021014
References:
 [1] N. Abdulaleem, $E$-invexity and generalized $E$-invexity in $E$-differentiable multiobjective programming, inITM Web of Conferences, EDP Sciences, 24 (2019), 01002. doi: 10.1051/itmconf/20192401002.  Google Scholar [2] N. Abdulaleem, $E$-optimality conditions for $E$-differentiable $E$-invex multiobjective programming problems, WSEAS Transactions on Mathematics, 18 (2019), 14-27.   Google Scholar [3] N. Abdulaleem, $E$-duality results for $E$-differentiable $E$-invex multiobjective programming problems, in Journal of Physics: Conference Series, IOP Publishing, 1294 (2019), 032027. doi: 10.1088/1742-6596/1294/3/032027.  Google Scholar [4] B. Aghezzaf, M. Hachimi, Generalized invexity and duality in multiobjective programming problems, Journal of Global Optimization, (2000), 91-101. doi: 10.1023/A:1008321026317.  Google Scholar [5] I. Ahmad, S. K. Gupta and A. Jayswal, On sufficiency and duality for nonsmooth multiobjective programming problems involving generalized $V$-$r$-invex functions, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 5920-5928.  doi: 10.1016/j.na.2011.05.058.  Google Scholar [6] T. Antczak and N. Abdulaleem, Optimality and duality results for $E$-differentiable multiobjective fractional programming problems under $E$-convexity, Journal of Inequalities and Applications, 2019 (2019), Article number: 292. doi: 10.1186/s13660-019-2237-x.  Google Scholar [7] T. Antczak and N. Abdulaleem, Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function, Journal of Industrial & Management Optimization, 16 (2020), 2971-2989.  doi: 10.3934/jimo.2019089.  Google Scholar [8] T. Antczak and N. Abdulaleem, $E$-optimality conditions and Wolfe $E$-duality for $E$-differentiable vector optimization problems with inequality and equality constraints, Journal of Nonlinear Sciences and Applications, 12 (2019), 745-764.  doi: 10.22436/jnsa.012.11.06.  Google Scholar [9] T. Antczak, $r$-preinvexity and $r$-invexity in mathematical programming, Computer and Mathematics with Applications, (2005), 551-566. doi: 10.1016/j.camwa.2005.01.024.  Google Scholar [10] T. Antczak, The notion of $V$-$r$-invexity in differentiable multiobjective programming, Journal of Applied Analysis, (2005), 63-79. doi: 10.1515/JAA.2005.63.  Google Scholar [11] T. Antczak, A class of $B-(p, r)$-invex functions and mathematical programming, Journal of Mathematical Analysis and Applications, 286 (2003), 187-206.  doi: 10.1016/S0022-247X(03)00469-4.  Google Scholar [12] T. Antczak, Optimality and duality for nonsmooth multiobjective programming problems with $V$-$r$-invexity, Journal of Global Optimization, 45 (2009), 319-334.  doi: 10.1007/s10898-008-9377-8.  Google Scholar [13] A. Ben-Israel and B. Mond, What is invexity?, Journal of the Australian Mathematical Society, 28 (1986), 1-9. doi: 10.1017/S0334270000005142.  Google Scholar [14] B. D. Craven and B. M. Glover, Invex functions and duality, Journal of the Australian Mathematical Society, 39 (1985), 1-20.  doi: 10.1017/S1446788700022126.  Google Scholar [15] M. A. Hanson and B. Mond, Further generalizations of convexity in mathematical programming, Journal of Information and Optimization Sciences, 3 (1982), 25-32.  doi: 10.1080/02522667.1982.10698716.  Google Scholar [16] M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Mathematical Programming, 37 (1987), 51-58.  doi: 10.1007/BF02591683.  Google Scholar [17] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.  Google Scholar [18] V. Jeyakumar and B. Mond, On generalised convex mathematical programming, The Anziam Journal, 34 (1992), 43-53.  doi: 10.1017/S0334270000007372.  Google Scholar [19] R. N. Kaul, S. K. Suneja and M. K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity, Journal of Optimization Theory and Applications, 80 (1994), 465-482.  doi: 10.1007/BF02207775.  Google Scholar [20] H. Kuk, G. M. Lee and D. S. Kim, Nonsmooth multiobjective programs with $V$-$\rho$-invexity, Indian Journal of Pure and Applied Mathematics, 29 (1998), 405-412.   Google Scholar [21] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.  Google Scholar [22] A. A. Megahed, H. G. Gomma, E. A. Youness and A. Z. El-Banna, Optimality conditions of $E$-convex programming for an $E$-differentiable function, Journal of Inequalities and Applications, 2013 (2013), Article number: 246. doi: 10.1186/1029-242X-2013-246.  Google Scholar [23] S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications, 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.  Google Scholar [24] B. Mond and T. Weir, Generalized concavity and duality, in Generalized Concavity in Optimization and Economics(eds. Schaible, W.T. Ziemba), Academic press, New York, (1981), 263-275.  Google Scholar [25] R. N. Mukherjee and S. K. Mishra, Sufficient optimality criteria and duality for multiobjective variational problems with $V$-invexity, Indian Journal of Pure and Applied Mathematics, 25 (1994), 801-813.   Google Scholar [26] G. R. Piao and L. Jiao, Optimality and mixed duality in multiobjective $E$-convex programming, Journal of Inequalities and Applications, 2015 (2015), 1-13.  doi: 10.1186/s13660-015-0854-6.  Google Scholar [27] V. Preda, I. Stancu-Minasian, M. Beldiman and A. M. Stancu, Generalized $V$-univexity type-I for multiobjective programming with $\eta$-set functions, Journal of Global Optimization, 44 (2009), Article number: 131. doi: 10.1007/s10898-008-9315-9.  Google Scholar [28] L. V. Reddy and R. N. Mukherjee, Composite nonsmooth multiobjective programs with $V$-$\rho$-invexity, Journal of Mathematical Analysis and Applications, 235 (1999), 567-577.  doi: 10.1006/jmaa.1999.6409.  Google Scholar [29] C. Singh, Optimality conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications, 53 (1987), 115-123.  doi: 10.1007/BF00938820.  Google Scholar [30] T. Weir and B. Mond, Preinvex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136 (1988), 29-38.  doi: 10.1016/0022-247X(88)90113-8.  Google Scholar [31] C. Yan and B. Feng, Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized $(F, \rho)$-$V$-Type I functions, Journal of Mathematical Modelling and Algorithms in Operations Research, 14 (2015), 159-172.  doi: 10.1007/s10852-014-9264-x.  Google Scholar [32] 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 [33] E. A. Youness and T. Emam, Characterization of efficient solutions for multi-objective optimization problems involving semi-strong and generalized semi-strong $E$-convexity, Acta Mathematica Scientia, 28 (2008), 7-16.  doi: 10.1016/S0252-9602(08)60002-8.  Google Scholar

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References:
 [1] N. Abdulaleem, $E$-invexity and generalized $E$-invexity in $E$-differentiable multiobjective programming, inITM Web of Conferences, EDP Sciences, 24 (2019), 01002. doi: 10.1051/itmconf/20192401002.  Google Scholar [2] N. Abdulaleem, $E$-optimality conditions for $E$-differentiable $E$-invex multiobjective programming problems, WSEAS Transactions on Mathematics, 18 (2019), 14-27.   Google Scholar [3] N. Abdulaleem, $E$-duality results for $E$-differentiable $E$-invex multiobjective programming problems, in Journal of Physics: Conference Series, IOP Publishing, 1294 (2019), 032027. doi: 10.1088/1742-6596/1294/3/032027.  Google Scholar [4] B. Aghezzaf, M. Hachimi, Generalized invexity and duality in multiobjective programming problems, Journal of Global Optimization, (2000), 91-101. doi: 10.1023/A:1008321026317.  Google Scholar [5] I. Ahmad, S. K. Gupta and A. Jayswal, On sufficiency and duality for nonsmooth multiobjective programming problems involving generalized $V$-$r$-invex functions, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 5920-5928.  doi: 10.1016/j.na.2011.05.058.  Google Scholar [6] T. Antczak and N. Abdulaleem, Optimality and duality results for $E$-differentiable multiobjective fractional programming problems under $E$-convexity, Journal of Inequalities and Applications, 2019 (2019), Article number: 292. doi: 10.1186/s13660-019-2237-x.  Google Scholar [7] T. Antczak and N. Abdulaleem, Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function, Journal of Industrial & Management Optimization, 16 (2020), 2971-2989.  doi: 10.3934/jimo.2019089.  Google Scholar [8] T. Antczak and N. Abdulaleem, $E$-optimality conditions and Wolfe $E$-duality for $E$-differentiable vector optimization problems with inequality and equality constraints, Journal of Nonlinear Sciences and Applications, 12 (2019), 745-764.  doi: 10.22436/jnsa.012.11.06.  Google Scholar [9] T. Antczak, $r$-preinvexity and $r$-invexity in mathematical programming, Computer and Mathematics with Applications, (2005), 551-566. doi: 10.1016/j.camwa.2005.01.024.  Google Scholar [10] T. Antczak, The notion of $V$-$r$-invexity in differentiable multiobjective programming, Journal of Applied Analysis, (2005), 63-79. doi: 10.1515/JAA.2005.63.  Google Scholar [11] T. Antczak, A class of $B-(p, r)$-invex functions and mathematical programming, Journal of Mathematical Analysis and Applications, 286 (2003), 187-206.  doi: 10.1016/S0022-247X(03)00469-4.  Google Scholar [12] T. Antczak, Optimality and duality for nonsmooth multiobjective programming problems with $V$-$r$-invexity, Journal of Global Optimization, 45 (2009), 319-334.  doi: 10.1007/s10898-008-9377-8.  Google Scholar [13] A. Ben-Israel and B. Mond, What is invexity?, Journal of the Australian Mathematical Society, 28 (1986), 1-9. doi: 10.1017/S0334270000005142.  Google Scholar [14] B. D. Craven and B. M. Glover, Invex functions and duality, Journal of the Australian Mathematical Society, 39 (1985), 1-20.  doi: 10.1017/S1446788700022126.  Google Scholar [15] M. A. Hanson and B. Mond, Further generalizations of convexity in mathematical programming, Journal of Information and Optimization Sciences, 3 (1982), 25-32.  doi: 10.1080/02522667.1982.10698716.  Google Scholar [16] M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Mathematical Programming, 37 (1987), 51-58.  doi: 10.1007/BF02591683.  Google Scholar [17] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.  Google Scholar [18] V. Jeyakumar and B. Mond, On generalised convex mathematical programming, The Anziam Journal, 34 (1992), 43-53.  doi: 10.1017/S0334270000007372.  Google Scholar [19] R. N. Kaul, S. K. Suneja and M. K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity, Journal of Optimization Theory and Applications, 80 (1994), 465-482.  doi: 10.1007/BF02207775.  Google Scholar [20] H. Kuk, G. M. Lee and D. S. Kim, Nonsmooth multiobjective programs with $V$-$\rho$-invexity, Indian Journal of Pure and Applied Mathematics, 29 (1998), 405-412.   Google Scholar [21] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.  Google Scholar [22] A. A. Megahed, H. G. Gomma, E. A. Youness and A. Z. El-Banna, Optimality conditions of $E$-convex programming for an $E$-differentiable function, Journal of Inequalities and Applications, 2013 (2013), Article number: 246. doi: 10.1186/1029-242X-2013-246.  Google Scholar [23] S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications, 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.  Google Scholar [24] B. Mond and T. Weir, Generalized concavity and duality, in Generalized Concavity in Optimization and Economics(eds. Schaible, W.T. Ziemba), Academic press, New York, (1981), 263-275.  Google Scholar [25] R. N. Mukherjee and S. K. Mishra, Sufficient optimality criteria and duality for multiobjective variational problems with $V$-invexity, Indian Journal of Pure and Applied Mathematics, 25 (1994), 801-813.   Google Scholar [26] G. R. Piao and L. Jiao, Optimality and mixed duality in multiobjective $E$-convex programming, Journal of Inequalities and Applications, 2015 (2015), 1-13.  doi: 10.1186/s13660-015-0854-6.  Google Scholar [27] V. Preda, I. Stancu-Minasian, M. Beldiman and A. M. Stancu, Generalized $V$-univexity type-I for multiobjective programming with $\eta$-set functions, Journal of Global Optimization, 44 (2009), Article number: 131. doi: 10.1007/s10898-008-9315-9.  Google Scholar [28] L. V. Reddy and R. N. Mukherjee, Composite nonsmooth multiobjective programs with $V$-$\rho$-invexity, Journal of Mathematical Analysis and Applications, 235 (1999), 567-577.  doi: 10.1006/jmaa.1999.6409.  Google Scholar [29] C. Singh, Optimality conditions in multiobjective differentiable programming, Journal of Optimization Theory and Applications, 53 (1987), 115-123.  doi: 10.1007/BF00938820.  Google Scholar [30] T. Weir and B. Mond, Preinvex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136 (1988), 29-38.  doi: 10.1016/0022-247X(88)90113-8.  Google Scholar [31] C. Yan and B. Feng, Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized $(F, \rho)$-$V$-Type I functions, Journal of Mathematical Modelling and Algorithms in Operations Research, 14 (2015), 159-172.  doi: 10.1007/s10852-014-9264-x.  Google Scholar [32] 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 [33] E. A. Youness and T. Emam, Characterization of efficient solutions for multi-objective optimization problems involving semi-strong and generalized semi-strong $E$-convexity, Acta Mathematica Scientia, 28 (2008), 7-16.  doi: 10.1016/S0252-9602(08)60002-8.  Google Scholar
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