Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions

doi:10.1007/s40305-021-00381-6

Journal of the Operations Research Society of China ›› 2023, Vol. 11 ›› Issue (3): 505-527.doi: 10.1007/s40305-021-00381-6

Previous Articles Next Articles

Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions

Izhar Ahmad¹, Krishna Kummari², S. Al-Homidan¹

1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia;
2. Department of Mathematics, School of Science, GITAM, Hyderabad Campus, Hyderabad, 502329, India

Received:2020-04-08 Revised:2021-10-30 Online:2023-09-30 Published:2023-09-07
Contact: Krishna Kummari, Izhar Ahmad, S. Al-Homidan E-mail:krishna.maths@gmail.com;drizhar@kfupm.edu.sa;homidan@kfupm.edu.sa

Abstract

Abstract: In this paper, a new concept of generalized-affineness type of functions is introduced. This class of functions is more general than some of the corresponding ones discussed in Chuong (Nonlinear Anal Theory Methods Appl 75:5044–5052, 2018), Sach et al. (J Global Optim 27:51–81, 2003) and Nobakhtian (Comput Math Appl 51:1385–1394, 2006). These concepts are used to discuss the sufficient optimality conditions for the interval-valued programming problem in terms of the limiting/Mordukhovich subdifferential of locally Lipschitz functions. Furthermore, two types of dual problems, namely Mond–Weir type and mixed type duals are formulated for an interval-valued programming problem and usual duality theorems are derived. Our results improve and generalize the results appeared in Kummari and Ahmad (UPB Sci Bull Ser A 82(1):45–54, 2020).

Key words: Mordukhovich subdifferential, Locally Lipschitz functions, Generalized invex-infine function, Interval-valued programming, LU-optimal, Constraint qualifications, Duality

CLC Number:

Izhar Ahmad, Krishna Kummari, S. Al-Homidan. Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions[J]. Journal of the Operations Research Society of China, 2023, 11(3): 505-527.

TrendMD

References

[1] Chuong, T.D.: L-invex-infine functions and applications. Nonlinear Anal. Theory Methods Appl. 75, 5044–5052(2018)
[2] Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)
[3] Ahmad, I., Jayswal, A., Banerjee, J.: On interval-valued optimization problems with generalized invex functions. J. Inequal. Appl. 2013, 313(2013)
[4] Ahmad, I., Kummari, K., Al-Homidan, S.: Sufficiency and duality for interval-valued optimization problems with vanishing constraints using weak constraint qualification. Int. J. Anal. Appl. 18(5), 784–798(2020)
[5] Kummari, K., Ahmad, I.: Sufficient optimality conditions and duality for nonsmooth interval-valued optimization problems via L-invex-infine functions. U.P.B. Sci. Bull. Ser. A 82(1), 45–54(2020)
[6] Kong, X., Zhang, Y., Yu, G.: Optimality and duality in set-valued optimization utilizing limit sets. Open Math. 16, 1128–1139(2018)
[7] Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A., Hernández-Jiménez, B.: Optimality and duality on Riemannian manifolds. Taiwan. J. Math. 22(5), 1245–1259(2018)
[8] Ahmad, I., Jayswal, A., Al-Homidan, S., Banerjee, J.: Sufficiency and duality in interval-valued variational programming. Neural Comput. Appl. 31, 4423–4433(2019)
[9] Luu, D.V., Mai, T.T.: Optimality and duality in constrained interval-valued optimization. 4OR-Q. J. Oper. Res. 16(3), 311–337(2018)
[10] Wu, H.: Solving the interval-valued optimization problems based on the concept of null set. J. Ind. Manag. Optim. 14(3), 1157–1178(2018)
[11] Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969(1976)
[12] Chuong, T.D., Kim, D.S.: Optimality conditions and duality in nonsmooth multiobjective optimization problems. Ann. Oper. Res. 217, 117–136(2014)
[13] Khanh, P.D., Yao, J., Yen, N.D.: The Mordukhovich subdifferentials and directions of descent. J. Optim. Theory Appl. 172, 518–534(2017)
[14] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
[15] Mordukhovich, B.S., Shao, Y.H.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348(4), 1235–1280(1996)
[16] Roshchina, V.: Mordukhovich subdifferential of pointwise minimum of approximate convex functions. Optim. Method Softw. 25, 129–141(2010)
[17] Sach, P.H., LEE, G.M., Kim, D.S.: Infine functions, nonsmooth alternative theorems and vector optimization problems. J. Global Optim. 27, 51–81(2003)
[18] Nobakhtian, S.: Infine functions and nonsmooth multiobjective optimization problems. Comput. Math. Appl. 51, 1385–1394(2006)
[19] Sun, Y., Wang, L.: Optimality conditions and duality in nondifferentiable interval-valued programming. J. Ind. Manag. Optim. 9, 131–142(2013)

Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 8

Recommended Articles

Metrics

Comments

[1]	Sheng-Long Hu. Certifying the Global Optimality of Quartic Minimization over the Sphere [J]. Journal of the Operations Research Society of China, 2022, 10(2): 241-287.
[2]	Promila Kumar, Jyoti Dagar. Optimality and Duality for Multiobjective Semi-infinite Variational Problem Using Higher-Order B-type I Functions [J]. Journal of the Operations Research Society of China, 2021, 9(2): 375-393.
[3]	Bhuwan Chandra Joshi, Shashi Kant Mishra, Pankaj Kumar. On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity [J]. Journal of the Operations Research Society of China, 2020, 8(4): 619-636.
[4]	Khushboo Verma, Pankaj Mathur, Tilak Raj Gulati. A New Approach on Mixed-Type Nondifferentiable Higher-Order Symmetric Duality [J]. Journal of the Operations Research Society of China, 2019, 7(2): 321-335.
[5]	Promila Kumar, Bharti Sharma, Jyoti Dagar. Interval-Valued Programming Problem with Infinite Constraints [J]. Journal of the Operations Research Society of China, 2018, 6(4): 611-626.
[6]	Murat Adivar, Shu-Cherng Fang. Convex Analysis and Duality over Discrete Domains [J]. Journal of the Operations Research Society of China, 2018, 6(2): 189-247.
[7]	Anurag Jayswal · Jonaki Banerjee. An Exact l1 Penalty Approach for Interval-ValuedProgramming Problem [J]. Journal of the Operations Research Society of China, 2016, 4(4): 461-.
[8]	Xin- Min Yang · Jin Yang · Tsz Leung Yip. On Nondifferentiable Higher-Order Symmetric Duality in Multiobjective Programming Involving Cones [J]. Journal of the Operations Research Society of China, 2013, 1(4): 453-466.