# American Institute of Mathematical Sciences

• Previous Article
Pricing decisions for closed-loop supply chains with technology licensing and carbon constraint under reward-penalty mechanism
• JIMO Home
• This Issue
• Next Article
Performance analysis and system optimization of an energy-saving mechanism in cloud computing with correlated traffic
doi: 10.3934/jimo.2022004
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.

## 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. Kanzi, J. 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. Kaul, S. 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. Kanzi, J. 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. Kaul, S. 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.
 [1] Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $E$-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089 [2] Najeeb Abdulaleem. $V$-$E$-invexity in $E$-differentiable multiobjective programming. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 427-443. doi: 10.3934/naco.2021014 [3] Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615 [4] 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 and Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081 [5] Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021216 [6] Baoxiang Wang. E-Besov spaces and dissipative equations. Communications on Pure and Applied Analysis, 2004, 3 (4) : 883-919. doi: 10.3934/cpaa.2004.3.883 [7] Tao Chen, Yunping Jiang, Gaofei Zhang. No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883 [8] Shunfu Jin, Wuyi Yue. Performance analysis and evaluation for power saving class type III in IEEE 802.16e network. Journal of Industrial and Management Optimization, 2010, 6 (3) : 691-708. doi: 10.3934/jimo.2010.6.691 [9] Frank Merle, Hatem Zaag. O.D.E. type behavior of blow-up solutions of nonlinear heat equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 435-450. doi: 10.3934/dcds.2002.8.435 [10] Akhlad Iqbal, Praveen Kumar. Geodesic $\mathcal{E}$-prequasi-invex function and its applications to non-linear programming problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021040 [11] Waldyr M. Oliva, Gláucio Terra. Improving E. Cartan considerations on the invariance of nonholonomic mechanics. Journal of Geometric Mechanics, 2019, 11 (3) : 439-446. doi: 10.3934/jgm.2019022 [12] Panayotis Smyrnelis. Connecting orbits in Hilbert spaces and applications to P.D.E. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2797-2818. doi: 10.3934/cpaa.2020122 [13] Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 643-653. doi: 10.3934/naco.2013.3.643 [14] Augusto Visintin. P.D.E.s with hysteresis 30 years later. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 793-816. doi: 10.3934/dcdss.2015.8.793 [15] M. Guru Prem Prasad, Tarakanta Nayak. Dynamics of { $\lambda tanh(e^z): \lambda \in R$\ ${ 0 }$ }. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 121-138. doi: 10.3934/dcds.2007.19.121 [16] 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 and Management Optimization, 2018, 14 (2) : 561-582. doi: 10.3934/jimo.2017060 [17] Vladimir V. Marchenko, Klavdii V. Maslov, Dmitry Shepelsky, V. V. Zhikov. E.Ya.Khruslov. On the occasion of his 70th birthday. Networks and Heterogeneous Media, 2008, 3 (3) : 647-650. doi: 10.3934/nhm.2008.3.647 [18] Fei Gao. Data encryption algorithm for e-commerce platform based on blockchain technology. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1457-1470. doi: 10.3934/dcdss.2019100 [19] 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 [20] Caili Sang, Zhen Chen. $E$-eigenvalue localization sets for tensors. Journal of Industrial and Management Optimization, 2020, 16 (4) : 2045-2063. doi: 10.3934/jimo.2019042

2021 Impact Factor: 1.411