Optimality in interval fractional programming problems using d-invexity

Document Type : Research Paper

Authors

1 Department of Mathematics, Sardar Vallabhbhai National Institute of Technology

2 Department of Mathematics, Sardar Vallabhbhai National Institute of Technology, Surat

Abstract

In this paper, an interval fractional optimization problem with directionally differentiable functions is considered, and the d-invexity concept is introduced for interval-valued functions. Slater's constraint qualification and pre-invex directional derivative assumption are used to establish the necessary optimality conditions. Further, sufficient optimality conditions are derived under the d-invexity assumption, considering the LU-solution concept. As an application of interval fractional problems, a portfolio optimization problem with uncertain return and risk parameters subject to interval liquidity constraints is considered, and an optimal solution is obtained using the results developed in this paper. Also, the portfolio optimization problem is solved using the proposed global criteria method for interval optimization problems and two methods available in the literature. Moreover, to check the efficiency of the proposed method, a comparison between different methods is presented. Throughout the paper, non-trivial examples are presented at appropriate places to provide a better understanding of the results developed.

Keywords

Main Subjects

References

[1] I. Ahmad, Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems,
Continuous Optimization and Variational Inequalities, 1 (2018), 71-94. https://doi.org/10.1201/9781003289883
[2] T. Antczak, Multiobjective programming under d-invexity, European Journal of Operational Research, 137(1) (2002),
28-36. https://doi.org/10.1016/S0377-2217(01)00092-3
[3] A. K. Bhurjee, G. Panda, Efficient solution of interval optimization problem, Mathematical Methods of Operations
Research, 76 (2012), 273-288. https://doi.org/10.1007/s00186-012-0399-0
[4] L. M. Boychuk, V. O. Ovchinnikov, Principal methods of solution of multicriterial optimization problems (survey),
Soviet Automatic Control, 6(3) (1973), 1-4.
[5] S. Chen, The KKT optimality conditions for optimization problem with interval-valued objective function on
Hadamard manifolds, Optimization, 71(3) (2022), 613-632. https://doi.org/10.1080/02331934.2020.1810248
[6] M. Ciontescu, S. Treant¸˘a, On some connections between interval-valued variational control problems and the associated inequalities, Results in Control and Optimization, 12 (2023), 100300. https://doi.org/10.1016/j.rico.
2023.100300
[7] B. A. Dar, A. Jayswal, D. Singh, Optimality, duality and saddle point analysis for interval-valued nondifferentiable
multiobjective fractional programming problems, Optimization, 70(5-6) (2021), 1275-1305. https://doi.org/10.
1080/02331934.2020.1819276
[8] I. P. Debnath, S. K. Gupta, The Karush-Kuhn-Tucker conditions for multiple objective fractional interval valued optimization problems, RAIRO-Operations Research, 54(4) (2020), 1161-1188. https://doi.org/10.1051/ro/2019055
[9] I. P. Debnath, S. K. Gupta, Necessary and sufficient optimality conditions for fractional interval-valued optimization
problems, Decision Science in Action: Theory and Applications of Modern Decision Analytic Optimisation, Springer
Singapore, (2021), 155-173. https://doi.org/10.1007/978-981-13-0860-4-12
[10] B. H. Faaland, L. N. Jacob, The linear fractional portfolio selection problem, Management Science, 27(12) (1981),
1383-1389. https://doi.org/10.1287/mnsc.27.12.1383
[11] Y. Guo, G. Ye, W. Liu, D. Zhao, S. Treant¸˘a, Optimality conditions and duality for a class of generalized convex
interval-valued optimization problems, Mathematics, 9(22) (2021), 2979. https://doi.org/10.3390/math9222979
[12] Y. Guo, G. Ye, W. Liu, D. Zhao, S. Treant¸˘a, On symmetric gH-derivative: Applications to dual interval-valued
optimization problems, Chaos, Solitons and Fractals, 158 (2022), 112068. https://doi.org/10.1016/j.chaos.
2022.112068
[13] Y. Guo, G. Ye, W. Liu, D. Zhao, S. Treant¸˘a, Solving nonsmooth interval optimization problems based on intervalvalued
symmetric invexity, Chaos, Solitons and Fractals, 174 (2023), 113834. https://doi.org/10.1016/j.chaos.
2023.113834
[14] M. B. Khan, J. E. Mac´ıas-D´ıaz, S. Treant¸˘a, M. S. Soliman, Some Fej´er-Type inequalities for generalized intervalvalued
convex functions, Mathematics, 10(20) (2022), 3851. https://doi.org/10.3390/math10203851
[15] M. B. Khan, J. E. Mac´ıas-D´ıaz, S. Treant¸˘a, M. S. Soliman, H. G. Zaini, Hermite-Hadamard inequalities in fractional
calculus for left and right harmonically convex functions via interval-valued settings, Fractal and Fractional, 6(4)
(2022), 178. https://doi.org/10.3390/fractalfract6040178
[16] M. B. Khan, G. Santos-Garc´ıa, H. Budak, S. Treant¸˘a, M. S. Soliman, Some new versions of Jensen, Schur
and Hermite-Hadamard type inequalities for (p, J)-convex fuzzy-interval-valued functions, AIMS Mathematics, 8(3)
(2023), 7437-7470. https://doi.org/10.3934/math.2023374
[17] P. Kumar, A. K. Bhurjee, An efficient solution of nonlinear enhanced interval optimization problems and
its application to portfolio optimization, Soft Computing, 25(7) (2021), 5423-5436. https://doi.org/10.1007/
s00500-020-05541-z
[18] P. Kumar, G. Panda, U. C. Gupta, Generalized quadratic programming problem with interval uncertainty, FUZZ
IEEE 2013. In:2013 IEEE International Conference on Fuzzy Systems, (2013), 1-7. https://doi.org/10.1109/
FUZZ-IEEE.2013.6622375
[19] J. C. T. Mao, Essentials of portfolio diversification strategy, Journal of Finance, 25(5) (1970), 1109-1121. https:
//doi.org/10.2307/2325582
[20] H. M. Markowitz, Portfolio selection, The Journal of Finance, 7(1) (1952), 77-91. https://doi.org/10.2307/
2975974
[21] S. K. Mishra, S. Y. Wang, K. K. Lai, Optimality and duality in nondifferentiable and multiobjective programming
under generalized d-invexity, Journal of Global Optimization, 29 (2004), 425-438. https://doi.org/10.1023/B:
JOGO.0000047912.69270.8c
[22] H. H. Nguyen, N. T. Hoang, V. T. Nguyen, Necessary optimality conditions for approximate Pareto efficient
solutions of nonsmooth fractional interval-valued multiobjective optimization problems, HPU2 Journal of Science:
Natural Sciences and Technology, 1(2) (2022), 81-90. https://doi.org/10.56764/hpu2.jos.2022.2.1.81-90
[23] R. Osuna-G´omez, B. Hern´andez-Jim´enez, Y. Chalco-Cano, G. Ruiz-Garz´on, New efficiency conditions for multiobjective interval-valued programming problems, Information Sciences, 420 (2017), 235-248. https://doi.org/10.
1016/j.ins.2017.08.022
[24] S. K. Sahoo, M. A. Latif, O. M. Alsalami, S. Treant¸˘a, W. Sudsutad, J. Kongson, Hermite-Hadamard, Fej´er and
Pachpatte-type integral inequalities for center-radius order interval-valued preinvex functions, Fractal and Fractional,
6(9) (2022), 506. https://doi.org/10.3390/fractalfract6090506
[25] M. Salukvadze, On the existence of solutions in problems of optimization under vector-valued criteria, Journal of
Optimization Theory and Applications, 13 (1974), 203-217. https://doi.org/10.1007/BF00935540
[26] W. F. Sharpe, A linear programming algorithm for mutual fund portfolio selection, Management Science, 13(7)
(1967), 499-510. https://doi.org/10.1287/mnsc.13.7.499
[27] S. Treant¸˘a, M. Ciontescu, On optimal control problems with generalized invariant convex interval-valued functionals,
Journal of Industrial and Management Optimization, 20(11) (2024), 3317-3336. https://doi.org/10.3934/jimo.
2024055
[28] S. Treant¸˘a, P. Mishra, B. B. Upadhyay, Minty variational principle for nonsmooth interval-valued vector optimization
problems on Hadamard manifolds, Mathematics, 10(3) (2022), 523. https://doi.org/10.3390/math10030523
[29] T. Truong, B. Schneider, L. Nguyen Le Toan Nhat, Diamond alpha differentiability of interval-valued functions
and its applicability to interval differential equations on time scales, Iranian Journal of Fuzzy Systems, 21(1) (2024),
143-158. https://doi.org/10.22111/ijfs.2024.45184.7977
[30] F. R. Villanueva, V. A. de Oliveira, T. M. Costa, Optimality conditions for interval valued optimization problems,
Fuzzy Sets and Systems, 454 (2023), 38-55. https://doi.org/10.1016/j.fss.2022.06.020
[31] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, Journal of Mathematical Analysis and
Applications, 136(1) (1988), 29-38.
[32] Y. L. Ye, d-invexity and optimality conditions, Journal of Mathematical Analysis and Applications, 162(1) (1991),
242-249. https://doi.org/10.1016/0022-247X(91)90190-B
[33] P. L. Yu, M. Zeleny, The set of all nondominated solutions in linear cases and a multicriteria simplex method, Journal
of Mathematical Analysis and Applications, 49(2) (1975), 430-468. https://doi.org/10.1016/0022-247X(75)
90189-4
Iranian Journal of Fuzzy Systems
Volume 22, Issue 1
January and February 2025
Pages 71-91

Files

Share

How to cite

Statistics