Duality and Optimality for Quasidifferentiable Interval-Valued Problems
Abstract
In this article, we explore the concept of interval-valued nonsmooth optimization problems using r-invexity in relation to convex compact sets. For the selected nonsmooth interval-valued problem (IP), we derive necessary and sufficient optimality criteria. In addition to that, we establish various duality theorems under r-invex quasidifferentiable with respect to a convex compact set that is equal to the Minkowski sum of their subdifferentials and superdifferentials. We draft a numerical example to support the results obtained in this paper. It is important to note that the Lagrange multipliers are nonconstant for the considered problem.
References
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T.W.Reiland, “Nonsmoothinvexity,” Bulletin of the Australian Mathematical Society, vol. 42, no. 3, pp. 437–446, 1990. doi: 10.1017/S0004972700028604
M.A.Hanson, “Onsufficiency of the kuhn-tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981. doi: 10.1016/0022-247X(81)90123-2
J. Cheng, W. Lu, Z. Liu, D. Wu, W. Gao, and J. Tan, “Robust optimization of engineering structures involving hybrid probabilistic and interval uncertainties,” Structural and Multidisciplinary Optimization, vol. 63, pp. 1327–1349, 2021. doi: 10.1007/s00158-020-02762-6
A. Karaaslan and M. Gezen, “The evaluation of renewable energy resources in turkey by integer multi-objective selection problem with interval coefficient,” Renewable Energy, vol. 182, pp. 842–854, 2022. doi: 10.1016/j.renene.2021.10.053
D.Yin, “Application of interval valued fuzzy linear programming for stock portfolio optimization,” Applied mathematics, vol. 9, no. 02, pp. 101–113, 2018. doi: 10.4236/am.2018.92007
H.-C. Wu, “The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function,” European Journal of operational research, vol. 176, no. 1, pp. 46–59, 2007. doi: 10.1016/j.ejor.2005.09.007
A. Jayswal, I. Stancu-Minasian, and I. Ahmad, “On sufficiency and duality for a class of interval-valued programming problems,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4119–4127, 2011. doi: 10.1016/j.amc.2011.09.041
A. K. Bhurjee and G. Panda, “Efficient solution of interval optimization problem,” Mathematical Methods of Operations Research, vol. 76, pp. 273–288, 2012. doi: 10.1007/s00186-0120399-0
I. Ahmad, A. Jayswal, and J. Banerjee, “On interval-valued optimization problems with generalized invex functions,” Journal of Inequalities and Applications, vol. 2013, no. 313, pp. 1–14, 2013. doi: 10.1186/1029-242X-2013-313
J. Zhang, S. Liu, L. Li, and Q. Feng, “The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function,” Optimization Letters, vol. 8, pp. 607–631, 2014. doi: 10.1007/s11590-012-0601-6
I. P. Debnath and N. Pokharna, “On optimality and duality in interval-valued variational problem with b-(p, r)-invexity,” RAIRO-Operations Research, vol. 55, pp. 1909–1932, 2021. doi: 10.1051/ro/2021088
F. Clarke, Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, 1983.
V. F. Dem’yanov and A. M. Rubinov, “On quasidifferentiable functionals,” Doklady Akademii Nauk SSSR, vol. 250, no. 1, pp. 21–25, 1980.
V. F. Dem’yanov and A. M. Rubinov, Quasidifferential calculus. Optimization Software Inc. Publications Division, New York, 1986.
Y. Gao, “Optimality conditions with Lagrange multipliers for inequality constrained quasidifferentiable optimization,” in Quasidifferentiability and Related Topics. Springer, Boston, MA, 2000, pp. 151–162. [Online]. Available: https://link.springer.com/chapter/10.1007/9781-4757-3137-8 6
L. Kuntz and S. Scholtes, “Constraint qualifications in quasidifferentiable optimization,” Mathematical Programming, vol. 60, pp. 339–347, 1993. doi: 10.1007/BF01580618
B. Luderer and R. R¨osiger, “On shapiro’s results in quasidifferential calculus,” Mathematical programming, vol. 46, pp. 403–407, 1990. doi: 10.1007/BF01585754
L. Polyakova, “On the minimization of a quasidifferentiable function subject to equality-type quasidifferentiable constraints,” in Quasidifferential Calculus. Springer, Berlin, Heidelberg, 1986, pp. 44–55. [Online]. Available: https://link.springer.com/chapter/10.1007/BFb0121135
A. Shapiro, “On optimality conditions in quasidifferentiable optimization,” SIAM Journal on Control and Optimization, vol. 22, no. 4, pp. 610–617, 1984. doi: 10.1137/0322037
A.Uderzo, “Quasi-multiplier rules for quasidifferentiable extremum problems,” Optimization, vol. 51, no. 6, pp. 761–795, 2002. doi: 10.1080/0233193021000015613
D. E. Ward, “A constraint qualification in quasidifferentiable programming,” Optimization, vol. 22, no. 5, pp. 661–668, 1991. doi: 10.1080/02331939108843709
T. Antczak, “Lipschitz r-invex functions and nonsmooth programming,” Numerical Functional Analysis and Optimization, vol. 23, no. 3-4, pp. 265–283, 2006. doi: 10.1081/NFA120006693
T. Antczak,“γ-preinvexity & γ-invexity in mathematical programming,” Computers and Mathematics with Applications, vol. 50, no. 3-4, pp. 551–566, 2005. doi: 10.1016/j.camwa.2005.01.024
T. Antczak,“Onoptimalityconditions and duality results in a class of nonconvex quasidifferentiable optimization problems,” Computational and Applied Mathematics, vol. 36, no. 3, pp. 12991314, 2017. doi: 10.1007/s40314-015-0283-7
H. N. Singh and V. Laha, “On quasidifferentiable multiobjective fractional programming,” Iranian Journal of Science and Technology, Transactions A: Science, vol. 46, no. 3, pp. 917925, 2022. doi: 10.1007/s40995-022-01309-2
H. N. Singh and V. Laha, “On minty variational principle for quasidifferentiable vector optimization problems,” Optimization Methods and Software, vol. 38, no. 2, pp. 243–261, 2023. doi: 10.1080/10556788.2022.2119235
A. K. Prasad, J. Khatri, and I. Ahmad, “Optimality conditions for an interval-valued vector problem,” Kybernetika, vol. 61, no. 2, pp. 221–237, 2025. doi: 10.14736/kyb-2025-2-0221
V. Laha, A. Dwivedi, and P. Jaiswal, “On quasidifferentiable interval-valued multiobjective optimization,” Annals of Operations Research, pp. 1–31, 2025. doi: 10.1007/s10479-02506627-3
T.W.Reiland, “Nonsmoothinvexity,” Bulletin of the Australian Mathematical Society, vol. 42, no. 3, pp. 437–446, 1990. doi: 10.1017/S0004972700028604
M.A.Hanson, “Onsufficiency of the kuhn-tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981. doi: 10.1016/0022-247X(81)90123-2
Published
2025-10-29
How to Cite
KUMAR PRASAD, Ashish; KHATRI, Julie; AHMAD, Izhar.
Duality and Optimality for Quasidifferentiable Interval-Valued Problems.
Yugoslav Journal of Operations Research, [S.l.], oct. 2025.
ISSN 2334-6043.
Available at: <https://yujor.fon.bg.ac.rs/index.php/yujor/article/view/1369>. Date accessed: 30 oct. 2025.
doi: https://doi.org/10.2298/YJOR241015035K.
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Research Articles
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