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doi: 10.3934/jimo.2022004
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Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions

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  May 2021 Revised  August 2021 Early access February 2022

In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable multiobjective programming problems with $ E $-differentiable functions. Namely, the concept of $ E $-type Ⅰ functions is defined for $ E $-differentiable multiobjective programming problem. Based on the introduced concept of generalized convexity, the sufficiency of the so-called $ E $-Karush–Kuhn–Tucker optimality conditions are established for a feasible point to be an $ E $-efficient or a weakly $ E $-efficient solution. Further, the so-called vector Mond-Weir $ E $-dual problem is defined for the considered $ E $-differentiable multiobjective programming problem and several $ E $-duality theorems in the sense of Mond-Weir are derived under appropriate generalized $ E $-type Ⅰ functions.

Citation: Najeeb Abdulaleem. Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022004
References:
[1]

N. Abdulaleem, $E$-invexity and generalized $E$-invexity in $E$-differentiable multiobjective programming, In ITM Web of Conferences, EDP Sciences, 24 (2019), 01002. doi: 10.1051/itmconf/20192401002.

[2]

N. Abdulaleem, $E$-optimality conditions for $E$-differentiable $E$-invex multiobjective programming problems, WSEAS Transactions on Mathematics, 18 (2019), 14-27. 

[3]

B. Aghezzaf and M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Global Optim., 18 (2000), 91-101.  doi: 10.1023/A:1008321026317.

[4]

T. Antczak and N. Abdulaleem, Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function, J. Ind. Manag. Optim., 16 (2020), 2971-2989.  doi: 10.3934/jimo.2019089.

[5]

T. Antczak and N. Abdulaleem, $E$-optimality conditions and Wolfe $E$-duality for $E$-differentiable vector optimization problems with inequality and equality constraints, J. Nonlinear Sci. Appl., 12 (2019), 745-764.  doi: 10.22436/jnsa.012.11.06.

[6]

T. Antczak and N. Abdulaleem, Optimality and duality results for $E$-differentiable multiobjective fractional programming problems under $E$-convexity, J. Inequal. Appl., 2019 (2019), Paper No. 292, 24 pp. doi: 10.1186/s13660-019-2237-x.

[7]

G. Caristi and N. Kanzi, Karush-Kuhn-Tuker type conditions for optimality of non-smooth multiobjective semi-infinite programming, International Journal of Mathematical Analysis, 9 (2015), 1929-1938.  doi: 10.12988/ijma.2015.56172.

[8]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.

[9]

M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Programming, 37 (1987), 51-58.  doi: 10.1007/BF02591683.

[10]

N. Kanzi, Karush-Kuhn-Tucker types optimality conditions for non-smooth semi-infinite vector optimization problems, J. Math. Ext., 9 (2015), 45-56. 

[11]

N. Kanzi, Necessary and sufficient conditions for (weakly) efficient of non-differentiable multi-objective semi-infinite programming problems, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 1537-1544.  doi: 10.1007/s40995-017-0156-6.

[12]

N. KanziJ. S. Ardekani and G. Caristi, Optimality, scalarization and duality in linear vector semi-infinite programming, Optimization, 67 (2018), 523-536.  doi: 10.1080/02331934.2018.1454921.

[13]

N. Kanzi and M. Soleimani-Damaneh, Characterization of the weakly efficient solutions in nonsmooth quasiconvex multiobjective optimization, J. Global Optim., 77 (2020), 627-641.  doi: 10.1007/s10898-020-00893-0.

[14]

R. N. KaulS. K. Suneja and M. K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity, J. Optim. Theory Appl., 80 (1994), 465-482.  doi: 10.1007/BF02207775.

[15]

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, J. Inequal. Appl., 2013 (2013), Article number: 246, 11 pp. doi: 10.1186/1029-242X-2013-246.

[16]

S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.

[17]

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.

[18]

N. G. Rueda and M. A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl., 130 (1988), 375-385.  doi: 10.1016/0022-247X(88)90313-7.

[19]

E. A. Youness, $E$-convex sets, $E$-convex functions and $E$-convex programming, J. Optim. Theory Appl., 102 (1999), 439-450.  doi: 10.1023/A:1021792726715.

show all references

References:
[1]

N. Abdulaleem, $E$-invexity and generalized $E$-invexity in $E$-differentiable multiobjective programming, In ITM Web of Conferences, EDP Sciences, 24 (2019), 01002. doi: 10.1051/itmconf/20192401002.

[2]

N. Abdulaleem, $E$-optimality conditions for $E$-differentiable $E$-invex multiobjective programming problems, WSEAS Transactions on Mathematics, 18 (2019), 14-27. 

[3]

B. Aghezzaf and M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Global Optim., 18 (2000), 91-101.  doi: 10.1023/A:1008321026317.

[4]

T. Antczak and N. Abdulaleem, Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function, J. Ind. Manag. Optim., 16 (2020), 2971-2989.  doi: 10.3934/jimo.2019089.

[5]

T. Antczak and N. Abdulaleem, $E$-optimality conditions and Wolfe $E$-duality for $E$-differentiable vector optimization problems with inequality and equality constraints, J. Nonlinear Sci. Appl., 12 (2019), 745-764.  doi: 10.22436/jnsa.012.11.06.

[6]

T. Antczak and N. Abdulaleem, Optimality and duality results for $E$-differentiable multiobjective fractional programming problems under $E$-convexity, J. Inequal. Appl., 2019 (2019), Paper No. 292, 24 pp. doi: 10.1186/s13660-019-2237-x.

[7]

G. Caristi and N. Kanzi, Karush-Kuhn-Tuker type conditions for optimality of non-smooth multiobjective semi-infinite programming, International Journal of Mathematical Analysis, 9 (2015), 1929-1938.  doi: 10.12988/ijma.2015.56172.

[8]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.

[9]

M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Programming, 37 (1987), 51-58.  doi: 10.1007/BF02591683.

[10]

N. Kanzi, Karush-Kuhn-Tucker types optimality conditions for non-smooth semi-infinite vector optimization problems, J. Math. Ext., 9 (2015), 45-56. 

[11]

N. Kanzi, Necessary and sufficient conditions for (weakly) efficient of non-differentiable multi-objective semi-infinite programming problems, Iran. J. Sci. Technol. Trans. A Sci., 42 (2018), 1537-1544.  doi: 10.1007/s40995-017-0156-6.

[12]

N. KanziJ. S. Ardekani and G. Caristi, Optimality, scalarization and duality in linear vector semi-infinite programming, Optimization, 67 (2018), 523-536.  doi: 10.1080/02331934.2018.1454921.

[13]

N. Kanzi and M. Soleimani-Damaneh, Characterization of the weakly efficient solutions in nonsmooth quasiconvex multiobjective optimization, J. Global Optim., 77 (2020), 627-641.  doi: 10.1007/s10898-020-00893-0.

[14]

R. N. KaulS. K. Suneja and M. K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving generalized invexity, J. Optim. Theory Appl., 80 (1994), 465-482.  doi: 10.1007/BF02207775.

[15]

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, J. Inequal. Appl., 2013 (2013), Article number: 246, 11 pp. doi: 10.1186/1029-242X-2013-246.

[16]

S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908.  doi: 10.1006/jmaa.1995.1057.

[17]

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.

[18]

N. G. Rueda and M. A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl., 130 (1988), 375-385.  doi: 10.1016/0022-247X(88)90313-7.

[19]

E. A. Youness, $E$-convex sets, $E$-convex functions and $E$-convex programming, J. Optim. Theory Appl., 102 (1999), 439-450.  doi: 10.1023/A:1021792726715.

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