International Journal of Nonlinear Analysis and Applications

A new class of generalized convex functions and mathematical programming

Document Type : Research Paper

Authors

1 Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

2 Department of Mathematics, Mahrah University, Al-Mahrah, Yemen

3 Department of Mathematics, Hadhramout University, Al-Mahrah, Yemen

4 Department of Mathematics, National Institute of Technology, Chaltlang, Aizawl, 796012, Mizoram, India

Abstract

In this paper, a new class of nonconvex optimization problem is considered, namely  $(h,\varphi)$-$(b,F,\rho)$-convexity is defined for $(h,\varphi)$-differentiable mathematical programming problem. The sufficiency of the so-called Karush-Kuhn-Tucker optimality conditions are established for the considered $(h,\varphi)$-differentiable mathematical programming problem under (generalized) $(h,\varphi)$-$(b,F,\rho)$-convexity hypotheses. Further, the so-called  Mond-Weir $(h,\varphi)$-dual problem is defined for the considered $(h,\varphi)$-differentiable mathematical programming problem and several duality theorems in the sense of Mond-Weir are derived under appropriate (generalized) $(h,\varphi)$-$(b,F,\rho)$-convex assumptions.

Keywords

[1] N. Abdulaleem, E-optimality conditions for E-differentiable E-invex multiobjective programming problems, WSEAS Trans. Math. 18 (2019), 14–27.
[2] N. Abdulaleem, E-B-invexity in E-differentiable mathematical programming, Results Control Optim. 4 (2021), 100046.
[3] N. Abdulaleem, V-E-invexity in E-differentiable multiobjective programming, Numer. Algebra Control Optim. 12 (2022), no. 2, 427.
[4] N. Abdulaleem, E-univex sets, E-univex functions and E-differentiable E-univex programming, Results Math. 78 (2023), no. 1, 3.
[5] N. Abdulaleem, Optimality conditions for a class of E-differentiable vector optimization problems with intervalvalued objective functions under E-invexity, Int. J. Comput. Math. 100 (2023), no. 7, 1601–1624.
[6] B. Aghezzaf and M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Glob. Optim. 18 (2000), 91–101.
[7] B. Aghezzaf and M. Hachimi, Sufficiency and duality in multiobjective programming involving generalized (F, ρ)-convexity, J. Math. Anal. Appl. 258 (2001), no. 2, 617–628.
[8] T. Antczak, S. Vinay, and B.S. Mohan, Optimality and duality results for (h,φ)-nondifferentiable multiobjective programming problems with (h,φ)-(b,F,ρ)-convex functions, Filomat 36 (2022), no. 12, 4139–4156.
[9] M. Avriel, Nonlinear Programming: Analysis and Methods, Englewood Cliffs, New Jersey: Prentice-Hall, 1976.
[10] A. Ben-Tal, On generalized means and generalized convex functions, J. Optim. Theory Appl. 21 (1977), 1–13.
[11] D. Bhatia and P. Jain, Generalized (F, ρ)-convexity and duality for non smooth multi-objective programs, Optimization 31 (1994), no. 2, 153–164.
[12] R.R. Egudo and B. Mond, Duality with generalized convexity, J. Aust. Math. Soc. Ser. B 28 (1986), 10–21.
[13] T.R. Gulati, Sufficiency and duality in multiobjective programming involving generalized F-convex functions, J. Math. Anal. Appl. 183 (1994), 181–195.
[14] M. Hachimi and B. Aghezzaf, Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions, J. Math. Anal. Appl. 296 (2004), 382–392.
[15] M. Hachimi and B. Aghezzaf, Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions, J. Math. Anal. Appl. 319 (2006), 110–123.
[16] M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545–550.
[17] M.A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program. 37 (1987), 51–58.
[18] M.A. Hanson and B. Mond, Further generalization of convexity in mathematical programming, J. Inf. Optim. Sci. 3 (1982), 25–32.
[19] S. Jha, P. Das, and S. Bandhyopadhyay, Characterization of LU-efficiency and saddle-point criteria for Fapproximated multiobjective interval-valued variational problems, Results Control Optim. 4 (2021), 100044.
[20] V. Jeyakumar, Strong and weak invexity in mathematical programming, Math. Oper. Res. 55 (1985), 109–125.
[21] R.N. Kaul, S.K. Suneja, and M.K. Srivastava, Optimality criteria and duality in multi objective optimization involving generalized invexity, J. Optim. Theory Appl. 80 (1994), 465–482.
[22] S.N. Majeed and A.A. Enad, On semi strongly (E,F)-convex functions and semi strongly (E,F)-convex optimization problems, J. Phys.: Conf. Ser. 1879 (2021), no. 2, 022110.
[23] S.K. Mishra, On multiple objective optimization with generalized uninvexity, J. Optim. Theory Appl. 224 (1998), 131–148.
[24] S.K. Mishra and M.A. Noor, Some nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 316 (2006), 472–482.
[25] S. Nobakhtian, Sufficiency in nonsmooth multiobjective programming involving generalized (F, ρ)-convexity, J. Optim. Theory Appl. 130 (2006), 359–365.
[26] P. Pandian, On sufficiency and duality for (b, F, ρ)-convex multiobjective programs, Indian J. Pure Appl. Math. 33 (2002), 463–473.
[27] V. Preda, On efficiency and duality for multiobjective programs, J. Math. Anal. Appl. 166 (1992), 65–377.
[28] N.G. Rueda and M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130 (1998), 375–385.
[29] N.G. Rueda, M.A. Hanson, and C. Singh, Optimality and duality with generalized convexity. J. Optim. Theory Appl. 86 (1995), 491–500.
[30] J.P. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983), 231–225.
[31] Y.H. Xu and S.Y. Liu, Necessary conditions for (h, φ)-mathematical programming, Int. Trans. Oper. Res. 6 (2002), no. 4, 21–30.
[32] Y.H. Xu and S.Y. Liu, Kuhn-Tucker necessary conditions for (h, φ)-multiobjective optimization problems, J. Syst. Sci. Complex. 17 (2004), 472–484.
[33] G. Yu and S. Liu, Optimality for (h, φ)-multiobjective programming involving generalized type-I functions, J. Glob. Optim. 41 (2008), 147–161.
[34] D. Yuan, A. Chinchuluun, X. Liu, and P.M. Pardalos, Generalized convexities and generalized gradients based on algebraic operations, J. Math. Anal. Appl. 321 (2006), 675–690.
[35] Q.X. Zhang, On sufficiency and duality of solutions for nonsmooth (h, φ)-semi-infinite programming, Acta Math. Appl. Sin. 24 (2001), 129–138.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 5
May 2025
Pages 153-164
  • Receive Date: 06 December 2023
  • Revise Date: 21 March 2024
  • Accept Date: 22 March 2024