The steepest descent method for fuzzy optimization problems under granular differentiability

Shexiang Hai; Liang He; Shexiang Hai; Liang He

doi:10.3934/math.2025463

AIMS Mathematics

2025, Volume 10, Issue 4: 10163-10186. doi: 10.3934/math.2025463

Previous Article Next Article

Research article

The steepest descent method for fuzzy optimization problems under granular differentiability

Shexiang Hai ^,,
Liang He

School of Science, Lanzhou University of Technology, Lanzhou 730050, China

Received: 26 January 2025 Revised: 17 April 2025 Accepted: 24 April 2025 Published: 28 April 2025
MSC : 03E72

This paper aimed to establish optimality conditions and develop an efficient solution method for unconstrained fuzzy optimization problems using the steepest descent method, based on the granular differentiability and granular convexity of fuzzy mappings. First, building upon the gr-difference for fuzzy numbers, the granular gradient, gr-differentiable and twice gr-differentiable of fuzzy mappings were researched. Furthermore, the efficient solution of fuzzy optimization was introduced under the granular convexity of fuzzy mappings. Finally, we proposed a steepest descent method for fuzzy optimization problems by employing the characterization function of fuzzy mappings. Our analysis demonstrated that the proposed method converged linearly under standard convexity conditions.
- fuzzy optimization,
- granular differentiability,
- efficient solution,
- steepest descent method
Citation: Shexiang Hai, Liang He. The steepest descent method for fuzzy optimization problems under granular differentiability[J]. AIMS Mathematics, 2025, 10(4): 10163-10186. doi: 10.3934/math.2025463

Related Papers:

Abstract

This paper aimed to establish optimality conditions and develop an efficient solution method for unconstrained fuzzy optimization problems using the steepest descent method, based on the granular differentiability and granular convexity of fuzzy mappings. First, building upon the gr-difference for fuzzy numbers, the granular gradient, gr-differentiable and twice gr-differentiable of fuzzy mappings were researched. Furthermore, the efficient solution of fuzzy optimization was introduced under the granular convexity of fuzzy mappings. Finally, we proposed a steepest descent method for fuzzy optimization problems by employing the characterization function of fuzzy mappings. Our analysis demonstrated that the proposed method converged linearly under standard convexity conditions.

References

[1]	R. E. Bellman, L. A. Zadeh, Decision-making in a fuzzy environment, Management Sci., 17 (1970), 141–164. https://doi.org/10.1287/mnsc.17.4.B141 doi: 10.1287/mnsc.17.4.B141
[2]	D. Qiu, C. X. Ouyang, Optimality conditions for fuzzy optimization in several variables under generalized differentiability, Fuzzy Sets Syst., 434 (2022), 1–19. https://doi.org/10.1016/j.fss.2021.05.006 doi: 10.1016/j.fss.2021.05.006
[3]	F. F. Shi, G. J. Ye, W. Liu, S. Treanţâ, Optimality and duality for nonconvex fuzzy optimization using granular differentiability method, Inf. Sci., 684 (2024), 121287. https://doi.org/10.1016/j.ins.2024.121287 doi: 10.1016/j.ins.2024.121287
[4]	M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983), 552–558. https://doi.org/10.1016/0022-247X(83)90169-5 doi: 10.1016/0022-247X(83)90169-5
[5]	G. X. Wang, C. X. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming, Fuzzy Sets Syst., 138 (2003), 559–591. https://doi.org/10.1016/S0165-0114(02)00440-2 doi: 10.1016/S0165-0114(02)00440-2
[6]	B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151 (2005), 581–599. https://doi.org/10.1016/j.fss.2004.08.001 doi: 10.1016/j.fss.2004.08.001
[7]	L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst., 161 (2010), 1564–1584. https://doi.org/10.1016/j.fss.2009.06.009 doi: 10.1016/j.fss.2009.06.009
[8]	L. T. Gomes, L. C. Barros, A note on the generalized difference and the generalized differentiability, Fuzzy Sets Syst., 280 (2015), 142–145. https://doi.org/10.1016/j.fss.2015.02.015 doi: 10.1016/j.fss.2015.02.015
[9]	B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119–141. https://doi.org/10.1016/j.fss.2012.10.003 doi: 10.1016/j.fss.2012.10.003
[10]	Y. Chalco-Cano, R. Rodríguez-López, M. Jiménez-Gamero, Characterizations of generalized differentiable fuzzy functions, Fuzzy Sets Syst., 295 (2016), 37–56. https://doi.org/10.1016/j.fss.2015.09.005 doi: 10.1016/j.fss.2015.09.005
[11]	S. X. Hai, Z. T. Gong, H. X. Li, Generalized differentiability for n-dimensional fuzzy-number-valued functions and fuzzy optimization, Inf. Sci., 374 (2016), 151–163. https://doi.org/10.1016/j.ins.2016.09.028 doi: 10.1016/j.ins.2016.09.028
[12]	M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Trans. Fuzzy Syst., 26 (2018), 310–323. https://doi.org/10.1109/TFUZZ.2017.2659731 doi: 10.1109/TFUZZ.2017.2659731
[13]	J. K. Zhang, X. Y. Chen, L. F. Li, X. J. Ma, Optimality conditions for fuzzy optimization problems under granular convexity concept, Fuzzy Sets Syst., 447 (2022), 54–75. https://doi.org/10.1016/j.fss.2022.01.004 doi: 10.1016/j.fss.2022.01.004
[14]	J. Y. Li, M. A. Noor, On properties of convex fuzzy mappings, Fuzzy Sets Syst., 219 (2013), 113–125. https://doi.org/10.1016/j.fss.2012.11.006 doi: 10.1016/j.fss.2012.11.006
[15]	L. F. Li, S. Y. Liu, J. K. Zhang, On fuzzy generalized convex mappings and optimality conditions for fuzzy weakly univex mappings, Fuzzy Sets Syst., 280 (2015), 107–132. https://doi.org/10.1016/j.fss.2015.02.007 doi: 10.1016/j.fss.2015.02.007
[16]	S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129–132. https://doi.org/10.1016/0165-0114(92)90256-4
[17]	N. Furukawa, Convexity and local Lipschitz continuity of fuzzy-valued mappings, Fuzzy Sets Syst., 93 (1998), 113–119. https://doi.org/10.1016/S0165-0114(96)00192-3 doi: 10.1016/S0165-0114(96)00192-3
[18]	Y. R. Syau, On convex and concave fuzzy mappings, Fuzzy Sets Syst., 103 (1999), 163–168. https://doi.org/10.1016/S0165-0114(97)00210-8 doi: 10.1016/S0165-0114(97)00210-8
[19]	C. Zhang, X. H. Yuan, E. S. Lee, Convex fuzzy mapping and operations of convex fuzzy mappings, Comput. Math. Appl., 51 (2006), 143–152. https://doi.org/10.1016/j.camwa.2004.12.019 doi: 10.1016/j.camwa.2004.12.019
[20]	B. Pang, F. G. Shi, Subcategories of the category of $L$-convex space, Fuzzy Sets Syst., 313 (2017), 61–74. https://doi.org/10.1016/j.fss.2016.02.014 doi: 10.1016/j.fss.2016.02.014
[21]	T. M. Costa, Y. C. Cano, R. O. Gomez, New preference order relationships and their application to multiobjective interval and fuzzy interval optimization problems, Fuzzy Sets Syst., 477 (2024), 108812. https://doi.org/10.1016/j.fss.2023.108812 doi: 10.1016/j.fss.2023.108812
[22]	T. Antczak, Optimality results for a class of nonconvex fuzzy optimization problems with granular differentiable objective functions, Fuzzy Sets Syst., 498 (2025), 109147. https://doi.org/10.1016/j.fss.2024.109147 doi: 10.1016/j.fss.2024.109147
[23]	U. M. Pirzada, V. D. Pathak, Newton method for solving the multi-variable fuzzy optimization problem, J. Optim. Theory Appl., 156 (2013), 867–881. https://doi.org/10.1007/s10957-012-0141-3 doi: 10.1007/s10957-012-0141-3
[24]	Y. C. Cano, G. N. Silva, A. R. Lizana, On the Newton method for solving fuzzy optimization problems, Fuzzy Sets Syst., 272 (2015), 60–69. https://doi.org/10.1016/j.fss.2015.02.001 doi: 10.1016/j.fss.2015.02.001
[25]	F. F. Shi, G. J. Ye, W. Liu, D. F. Zhao, A class of nonconvex fuzzy optimization problems under granular differentiability concept, Math. Comput. Simul., 211 (2023), 430–444. https://doi.org/10.1016/j.matcom.2023.04.021 doi: 10.1016/j.matcom.2023.04.021
[26]	H. Vu, N. V. Hoa, Uncertain fractional differential equations on a time scale under granular differentiability concept, Comput. Appl. Math., 38 (2019), 110. https://doi.org/10.1007/s40314-019-0873-x doi: 10.1007/s40314-019-0873-x
[27]	M. Najariyan, N. Pariz, H. Vu, Fuzzy linear singular differential equations under granular differentiability concept, Fuzzy Sets Syst., 429 (2022), 169–187. https://doi.org/10.1016/j.fss.2021.01.003 doi: 10.1016/j.fss.2021.01.003
[28]	B. Li, P. Afkhami, R. Khayamim, Z. Elmi, R. Moses, J. Sobanjo, et al., A holistic optimization-based approach for sustainable selection of level crossings for closure with safety, economic, and environmental considerations, Reliab. Eng. Syst. Saf., 249 (2024), 110197. https://doi.org/10.1016/j.ress.2024.110197 doi: 10.1016/j.ress.2024.110197

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)