Sufficiency and duality for E-differentiable vector optimization problems under generalized convexity

Year 2025, Volume: 54 Issue: 1, 125 - 141, 28.02.2025

N. Abdulaleem , Jınxia Cen , Koushik Das

Abstract

In this paper, a new class of nonconvex vector optimization problems is considered. The concepts of $E$-$B$-pseudoinvexity and $E$-$B$-quasiinvexity are introduced for $E$-differentiable functions. Then, the sufficiency of the so-called $E$-Karush-Kuhn-Tucker optimality conditions is established for the considered $E$-differentiable vector optimization problems under (generalized) $E$-$B$-invexity. To illustrate the aforesaid results, a nonsmooth vector programming problem with $E$-differentiable functions is studied. For the $E$-differentiable vector optimization problem, the so-called vector Mond-Weir $E$-dual problem is defined, and several $E$-dual theorems are established under (generalized) $E$-$B$-invexity hypotheses.

Keywords

$E$-differentiable function, $E$-$B$-invex function, generalized $E$-$B$-invexity, $E$-optimality condition, Mond-Weir $E$-duality

References

• [1] N. Abdulaleem, E-invexity and generalized E-invexity in E-differentiable multiobjective programming, ITM Web of Conferences, EDP Sciences 24, 01002, 2019.
• [2] N. Abdulaleem, E-optimality conditions for E-differentiable E-invex multiobjective programming problems, WSEAS Trans. Math. 18, 14–27, 2019.
• [3] N. Abdulaleem, E-B-invexity in E-differentiable mathematical programming, Results Control Optim. 4, 100046, 2021.
• [4] N. Abdulaleem, V -E-invexity in E-differentiable multiobjective programming, Numer. Algebra Control Optim. 12, 427-443, 2022.
• [5] N. Abdulaleem, Optimality conditions for a class of E-differentiable vector optimization problems with interval-valued objective functions under E-invexity, Int. J. Comput. Math 7 (100), 1601-1624, 2023.
• [6] N. Abdulaleem, Optimality and duality for E-differentiable multiobjective programming problems involving E-type I functions, J. Ind. Manag. Optim. 19, 1513–1527, 2023.
• [7] T. Antczak, (p, r)-invex sets and functions, Aust. J. Math. Anal. Appl. 263, 355–379, 2001.
• [8] T. Antczak, B-(p, r)-pre-invex functions, Folia Math Acta Un Lodziensis 11, 3-15, 2004.
• [9] T. Antczak, Generalized B-(p,r)-invexity functions and nonlinear mathematical programming, Numer. Funct. Anal. Optim. 30, 1–22, 2009.
• [10] T. Antczak and N. Abdulaleem, E-optimality conditions and Wolfe E-duality for Edifferentiable vector optimization problems with inequality and equality constraints, J. Nonlinear Sci. Appl. 12, 745–764, 2019.
• [11] CR. Bector, SK. Suneja and CS. Lalitha, Generalized B-vex functions and generalized B-vex programming, J. Optim. Theory Appl. 76, 561–576, 1993.
• [12] CR Bector and C. Singh, B-vex functions, J. Optim. Theory Appl. 71, 237–253, 1991.
• [13] A. Ben-Israel and B. Mond, What is invexity?, ANZIAM J. 28, 1–9, 1986.
• [14] C. Fulga and V. Preda, Nonlinear programming with E-preinvex and local E-preinvex functions, European J. Oper. Res. 192, 737–743, 2009.
• [15] MA. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80, 545–550, 1981.
• [16] MA. Hanson and B. Mond, Further generalizations of convexity in mathematical programming, J. Inf. Optim. Sci. 3, 25–32, 1982.
• [17] MA. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program. 37, 51–58, 1987.
• [18] V. Jeyakumar and B. Mond, On generalised convex mathematical programming, The ANZIAM Journal 34, 43–53, 1992.
• [19] N. Kanzi, Necessary and sufficient conditions for (weakly) efficient of nondifferentiable multiobjective semi-infinite programming problems, Iran. J. Sci. Technol. Trans. A: Sci. 42, 1537–1544, 2018.
• [20] N. Kanzi and M. Soleimani-Damaneh, Characterization of the weakly efficient solutions in nonsmooth quasiconvex multiobjective optimization, J. Global Optim. 77, 627–641, 2020.
• [21] AA. Megahed, HG. Gomma, EA. Youness and AZ. El-Banna, Optimality conditions of E-convex programming for an E-differentiable function, J. Inequal. Appl. 2013, 246, 2013.
• [22] SR. Mohan and SK. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl. 189, 901–908, 1995.
• [23] B. Mond and T. Weir, Generalized concavity and duality. Schaible, S., Ziemba, W. T. (Eds), Generalized Concavity in Optimization and Economics, 263–276, Academic Press, New York, 1981.
• [24] NG. Rueda and MA. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130, 375–385, 1988.
• [25] SK. Suneja, C. Singh and CR. Bector, Generalization of preinvex and B-vex functions, J. Optim. Theory Appl. 76, 577–587, 1993.
• [26] YR. Syau and ES. Lee, Some properties of E-convex functions, Appl. Math. Lett. 18, 1074–1080, 2005.
• [27] YR. Syau and ES. Lee, Generalizations of E-convex and B-vex functions, Comput. Math. Appl. 58, 711–716, 2009.
• [28] XM. Yang, On E-convex sets, E-convex functions and E-convex programming, J. Optim. Theory Appl. 109, 699–704, 2001.
• [29] EA. Youness, E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl. 102, 439–450, 1999.
• [30] EA. Youness, Optimality criteria in E-convex programming, Chaos Solit. Fractals 12, 1737–1745, 2001.