On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

Rohit Kumar Bhardwaj; Tirth Ram

doi:10.3934/mfc.2023036

Article Contents

2025, Volume 8, Issue 1: 36-49. Doi: 10.3934/mfc.2023036

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On interval-valued vector variational-like inequalities and vector optimization problems with generalized approximate invexity via convexificators

Rohit Kumar Bhardwaj and
Tirth Ram^,

Department of Mathematics, University of Jammu, Jammu-180006, India

^*Corresponding author: Tirth Ram
^*Corresponding author: Tirth Ram

Received: June 2023

Revised: August 2023

Early access: August 2023

Published: February 2025

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Abstract

In this paper, we establish the relationships between a class of interval-valued vector optimization problems and interval-valued vector variational-like inequality problems of both Stampacchia and Minty kinds in terms of convexificators. We also provide necessary and sufficient optimality requirements for locally strong quasi and approximately efficient solutions by using the concept of approximate LU-$ (\eta, e) $-invexity. Numerical example is also presented to validate the main result. Our newly proved results generalize some well-known results in the literature.

Keywords:

Convexificator,
efficient solutions,
quasi efficient solution,
interval-valued vector optimization,
vector variational-like inequality problems.

Mathematics Subject Classification: Primary: 49J52; Secondary: 58E17.

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References

[1]	Q. H. Ansari, E. Kobis and J. C. Yao, Vector Variational Inequalities and Vector Optimization. Theory and Applications, Vector Optim., Springer, Cham, 2018. doi: 10.1007/978-3-319-63049-6.
[2]	F. H. Clarke, Optimization and Nonsmooth Analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York, 1983.
[3]	S. Dafermos, Exchange price equilibria and variational inequalities, Math. Program, 46 (1990), 391-402. doi: 10.1007/BF01585753.
[4]	V. F. Demyanov, Convexification and Concavification of Positively Homogeneous Functions by the Same Family of Linear Functions, Report 3.208.802., Universita di Pisa, 1994.
[5]	F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, ariational Inequality and Complementarity Problems, Wiley-Intersci. Publ. John Wiley & Sons, Ltd., Chichester, (1980), 151-186.
[6]	M. Golestani and S. Nobakhtian, Convexificator and strong Kuhn-Tucker conditions, Comput. Math. Appl., 64 (2012), 550-557. doi: 10.1016/j.camwa.2011.12.047.
[7]	P. Gupta and S. K. Mishra, On minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity, A Journal Of Mathematical Programming and Operations Research, 67 (2018), 1157-1167. doi: 10.1080/02331934.2018.1466884.
[8]	D. I. Inoan, Existence results for systems of quasi-variational relations, Constr. Math. Anal., 2 (2019), 217-222. doi: 10.33205/cma.643397.
[9]	M. Jennane, E. M. Kalmoun and L. E. Fadil, Interval-valued vector optimization problems involving generalized approximate convexity, J. Math. Computer Sci., 26 (2020), 67-79. doi: 10.22436/jmcs.026.01.06.
[10]	V. Jeyakumar and D. T. Luc, Nonsmooth calculus, minimality, and monotonicity of convexificators, J. Optim. Theory Appl., 101 (1999), 599-621. doi: 10.1023/A:1021790120780.
[11]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math., 88, Academic Press, Inc., New York-London, 1980.
[12]	F. A. Khan, R. K. Bhardwaj, T. Ram and M. A. S. Tom, On approximate vector variational inequalities and vector optimization problem using convexificator, AIMS Mathematics, 7 (2022), 18870-18882. doi: 10.3934/math.20221039.
[13]	V. Laha and S. K. Mishra, On vector optimization problems and vector variational inequalities using convexificators, Optim., 66 (2017), 1837-1850. doi: 10.1080/02331934.2016.1250268.
[14]	R. Liu and R. Xu, Hermite-Hadamard type inequalities for harmonical $(h_{1}, h_{2})$-convex interval-valued functions, Mathematical Foundations of Computing, 4 (2021), 89-103. doi: 10.3934/mfc.2021005.
[15]	X. J. Long and N. J. Huang, Optimality conditions for efficiency on nonsmooth multiobjective programming problems, Taiwanese J. Math., 18 (2014), 687-699. doi: 10.11650/tjm.18.2014.3730.
[16]	J. Lu, L. Zhu and W. Gao, Remarks on bipolar cubic fuzzy graphs and its chemical applications, International Journal of Mathematics and Computer in Engineering, 1 (2023), 1-10. doi: 10.2478/ijmce-2023-0001.
[17]	D. V. Luu, Convexifcators and necessary conditions for efficiency, Optim., 63 (2013), 321-335. doi: 10.1080/02331934.2011.648636.
[18]	D. V. Luu, Necessary and sufficient conditions for efficiency via convexificators, J Optim Theory Appl., 160 (2014), 510-526. doi: 10.1007/s10957-013-0377-6.
[19]	P. Michel and J. P. Penot, A generalized derivative for calm and stable functions, Differ. Integral Equ., 5 (1992), 433-454.
[20]	S. K. Mishra and S. Y. Wang, Vector variational-like inequalities and non-smooth vector optimization problems, Nonlinear Analysis, 64 (2006), 1939-1945. doi: 10.1016/j.na.2005.07.030.
[21]	R. E. Moore, Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1966.
[22]	R. E. Moore, Method and Applications of Interval Analysis, SIAM Stud. Appl. Math., 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979.
[23]	B. S. Mordukhovich and Y. H. Shao, On nonconvex subdifferential calculus in Banach spaces, J. Convex Anal., 2 (1995), 211-227.
[24]	E. R. Nwaeze, Set inclusions of the Hermite-hadamard Type for m polynomial harmonically convex interval valued functions, Constr. Math. Anal., 4 (2021), 260-273. doi: 10.33205/cma.793456.
[25]	R. Osuna-Gomez, B. Hernadez-Jimenez and Y. Chalco-Cano, New efficiency conditions for multiobjective interval-valued programming problems, Inf Sci., 420 (2017), 235-248. doi: 10.1016/j.ins.2017.08.022.
[26]	A. D. Singh, B. A. Dar and D. S. Kim, Sufficiency and duality in non-smooth interval valued programming problems, J. Ind. Manag. Optim., 15 (2019), 647-665. doi: 10.3934/jimo.2018063.
[27]	B. B. Upadhyay and P. Mishra, On Minty variational principle for nonsmooth interval-valued multiobjective programming problems, Indo-French Seminar on Optimization, Variational Analysis and Applications, Springer Proc. Math. Stat., Springer, Singapore, 355 (2020), 265-282.
[28]	B. B. Upadhyay, I. M. Stancu-Minasian and P. Mishra, On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities, Optim., (2022), 1-5. doi: 10.1080/02331934.2022.2069569.
[29]	L. Y. Wei, Z. Yu and D. X. Zhou, Federated learning for minimizing nonsmooth convex loss functions, Mathematical Foundations of Computing, 4 (2023), 753-770. doi: 10.3934/mfc.2023026.
[30]	J. K. Zhang, Q. H. Zheng, X. J. Ma and L. F. Li, Relationships between interval-valued vector optimization problems and vector variational inequalities, Fuzzy Optim. Decis. Making, 15 (2016), 33-55. doi: 10.1007/s10700-015-9212-x.
[31]	R. X. Zhou, Z. B. Yang, M. Yu and D. A. Ralescu, A portfolio optimization model based on information entropy and fuzzy time series, Fuzzy Optim. Decis. Making, 14 (2015), 381-397. doi: 10.1007/s10700-015-9206-8.