Optimality conditions for multi-objective interval-valued E-convex functions with the use of $gH$-symmetrical differentiation

Document Type : Research Paper


1 Department of Mathematics, Hindu college, M.J.P. Rohilkhand University, Bareilly-243003, UP, India

2 Department of Mathematics, Aligarh Muslim University, Aligarh-202002, UP, India


In this paper, we introduce and discuss multi-objective interval-valued E-convex programming using gH-symmetrical differentiability. We prove nonlinear optimality conditions of Fritz John type for this context and construct an example to verify our results. Furthermore, we define LU-sE-pseudo convexity and LU-sE-quasi convexity for interval-valued functions and study some of their properties.


[1] I. Ahmad, D. Singh and B. Ahmad, Optimality conditions for invex interval valued nonlinear programming problems involving generalized H-derivative, Filomat 30 (2016), no. 8, 2121–2138.
[2] G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press. NY. 1983.
[3] T.Q. Bao and B.S. Mordukhovich, Set-valued optimization in welfare economics. Adv. Math. Econ. 13 (2010),
[4] Y. Bao, B. Zao and E. Bai, Directional differentiability of interval-valued functions, J. Math. Comput. Sci. 16
(2016), no. 4, 507–515.
[5] M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd Edition,
Willey publication, 2006
[6] D.P. Bertsekas, A. Nedic and A.E. Ozdaglar, Convex Analysis and Optimization, Athena Scientific. Belmont.
U.S.A. 2003.
[7] G.R. Bitran, Linear multiple objective problems with interval coefficients , Manag. Sci. 26 (1980), 694–706.
[8] Y. Chalco-Cano, W.A. Lodwick and A. Rufian-Lizana, Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optim. Decis. Mak. 12 (2013), 305–
[9] S. Chanas and D. Kuchta, Mutiobjective programming in optimization of interval objective functions-A generalized
approach, Eur. J. Oper. Res. 94 (1996), 594–598.
[10] B.D. Chung, T. Yao, C. Xie and A. Thorsen, Robust Optimization Model for a dynamic Network Design problem
Under Demand Uncertainty, Netw. Spat. Econ. 11 (2010), 371–389.
[11] I.P. Devnath and S.K. Gupta, The Karush-Kuhn-Tucker conditions for multiple objective fractional interval valued
optimization problems, Rairo-oper. Res. 54 (2020), 1161–1188.
[12] Y. Guo, Ye. Guoju, D. Zhao and W. Liu, gH-symmetrically derivative of interval-valued functions and application
in interval-valued optimization, Symmetry 11 (2019), 1203.
[13] M. Ida, Multiple objective linear programming with interval coefficients and its all efficient solutions, Proc. 35th
IEEE Conf. Decis. Control, Kobe, Japan, 13 December 1996, Volume 2, pp. 1247–1249.
[14] H. Ishibpchi and H.Tanaka, Multiobjective programming in optimization of interval valued objective functions,
Eur. J. Oper. Res. 48 (1990), 219–225.
[15] A. Jayswal, I.M. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems , Appl. Math. Comput. 218 (2011), 4119–4127.
[16] R.A. Minch, Applications of symmetric derivatives in mathematical programming, Math. Program. 1 (1971),
[17] R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
[18] R.E. Moore, Method and Application of Interval Analysis, SIAM, Philadelphia, 1979.
[19] G.M. Ostrovsky, Y.M. Volin and D.V. Golovashkin, Optimization problem of complex system under uncertainty,
Comput. Chem. Eng. 22 (1998), 1007–1015.
[20] S. Rastogi, A. Iqbal and S. Rajan, Optimality Conditions for E-convex Interval-valued Programming Problem
using gH-Symmetrical Derivative, submitted.
[21] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York. 1993.
[22] D. Singh, B.A. Dar and D.S. Kim, KKT optimality conditions in interval valued multiobjective programming with
generalized differentiable functions, Eur. J. Oper. Res. 254 (2016), 29–39.
[23] L. Stefanini and B. Bede, Generalized Hpkphara differentiability of interval valped fpnctions and interval differential eqpations , Nonlinear Anal, 71 (2009), 1311–1328.
[24] J. Tao and Z.H. Zhang, Properties of intervel vector valued arithmetic based on gH-difference, Math. Comput. 4(2015), 7–12.
[25] B.S. Thomson, Symmetric Properties of Real Functions, Dekker: New York. USA. 1994.
[26] H.C. Wu, The karush Kuhn tuker optimality conditions in an optimization problem with interval valued objective
functions, Eur. J. Oper. Res. 176 (2007), 46–59.
[27] H.C. Wu, On Interval valued Nonlinear Programming Problems, J. Math. Anal. Appl. 338 (2008), 299–316.
[28] H.C. Wu, The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued
objective functions , Fuzzy Optim. Decis. Mak. 8 (2009), 295–321.
[29] X.M. Yang. On E-convex set,E-convex functions and E-convex programming, J. Optim. Theory Appl. 109 (2001),
no. 3, 699–704.
[30] E.A Youness, E-convex set, E-convex functions and E-convex programming, J. Optim. Theory Appl. 102 (1999),
no. 2, 439–450.
[31] E.A. Youness. Optimality criteria in E-convex programming, Chaos Solitons Fractals 12 (2001), 1737–1745.
Volume 13, Issue 2
July 2022
Pages 3261-3270
  • Receive Date: 24 September 2021
  • Revise Date: 13 June 2022
  • Accept Date: 02 August 2022