Higher-Order Mond-Weir Duality of Set-Valued Fractional Minimax Programming Problems
Abstract
In this paper, we consider a set-valued fractional minimax programming problem (abbreviated as SVFMPP) (MFP), in which both the objective and constraint maps are set-valued. We use the concept of higher-order α-cone arcwisely connectivity, introduced by Das [1], as a generalization of higher-order cone arcwisely connected setvalued maps. We explore the higher-order Mond-Weir (MWD) form of duality based on the supposition of higher-order α-cone arcwisely connectivity and prove the associated higher-order converse, strong, and weak theorems of duality between the primary (MFP) and the analogous dual problem (MWD).
References
K. Das, “Set-valued parametric optimization problems with higher-order ρ-cone arcwise connectedness,” SeMA Journal, pp. 1–19, 2024. doi: 10.1007/s40324-023-00344-2
S. R. Yadav and R. N. Mukherjee, “Duality for fractional minimax programming problems,” Journal of the Australian Mathematical Society (Series B), vol. 31, no. 4, pp. 484–492, 1990. doi: 10.1017/S0334270000006809
S. Chandra and V. Kumar, “Duality in fractional minimax programming,” Journal of the Australian Mathematical Society (Series A), vol. 58, no. 3, pp. 376–386, 1995. doi: 10.1017/S1446788700038362
T. Weir, “Pseudoconvex minimax programming,” Utilitas Mathematica, vol. 42, pp. 234–240, 1992.
C. R. Bector and B. L. Bhatia, “Sufficient optimality conditions and duality for a minmax problem,” Utilitas Mathematica, vol. 27, pp. 229–247, 1985.
G. J. Zalmai, “Optimality criteria and duality for a class of minimax programming problems with generalized invexity conditions,” Utilitas Mathematica, vol. 32, pp. 35–57, 1987.
J. C. Liu and C. S. Wu, “On minimax fractional optimality conditions with (F,ρ)-convexity,” Journal of Mathematical Analysis and Applications, vol. 219, no. 1, pp. 36–51, 1998. doi: 10.1006/jmaa.1997.5785
I. Ahmad, “Optimality conditions and duality in fractional minimax programming involving generalized ρ-invexity,” Journal of Management Information Systems, vol. 19, pp. 165–180, 2003.
Z. A. Liang and Z. W. Shi, “Optimality conditions and duality for minimax fractional programming with generalized convexity,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 474–488, 2003. doi: 10.1016/S0022-247X(02)00553-X
H. C. Lai, J. C. Liu, and K. Tanaka, “Necessary and sufficient conditions for minimax fractional programming,” Journal of Mathematical Analysis and Applications, vol. 230, no. 2, pp. 311–328, 1999. doi: 10.1006/jmaa.1998.6204
H. C. Lai and J. C. Lee, “On duality theorems for nondifferentiable minimax fractional programming,” Journal of Computational and Applied Mathematics, vol. 146, no. 1, pp. 115–126, 2002. doi: 10.1016/S0377-0427(02)00422-3
I. Ahmad and Z. Husain, “Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity,” Journal of Optimization Theory and Applications, vol. 129, no. 2, pp. 255–275, 2006. doi: 10.1007/s10957-006-9057-0
S. J. Li, K. L. Teo, and X. Q. Yang, “Higher-order Mond-Weir duality for set-valued optimization,” Journal of Computational and Applied Mathematics, vol. 217, no. 2, pp. 339–349, 2008. doi: 10.1016/j.cam.2007.02.011
S. J. Li, K. L. Teo, and X. Q. Yang,, “Higher-order optimality conditions for set-valued optimization,” Journal of Optimization Theory and Applications, vol. 137, no. 3, pp. 533–553, 2008. doi: 10.1007/s10957-0079345-3
M. Avriel, Nonlinear Programming: Theory and Method. Englewood Cliffs, New Jersey: Prentice-Hall, 1976.
J. Y. Fu and Y. H. Wang, “Arcwise connected cone-convex functions and mathematical programming,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 339–352, 2003. doi: 10.1023/A:1025451422581
C. S. Lalitha, J. Dutta, and M. G. Govil, “Optimality criteria in set-valued optimization,” Journal of the Australian Mathematical Society, vol. 75, no. 2, pp. 221–232, 2003. doi: 10.1017/S1446788700003736
K.DasandC.Nahak,“Sufficiencyanddualityofset-valued optimization problems via higherorder contingent derivative,” Journal of Advanced Mathematical Studies, vol. 8, no. 1, pp. 137–151, 2015.
J. P. Aubin, Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions. Paris: Universit´e Paris IX-Dauphine, Centre de recherche de math´ematiques de la d´ecision, 1980.
J. P. Aubin and H. Frankowska, Set-Valued Analysis. Boston: Birh¨auser, 1990.
J. Jahn and R. Rauh, “Contingent epiderivatives and set-valued optimization,” Mathematical Methods of Operations Research, vol. 46, no. 2, pp. 193–211, 1997. doi: 10.1007/BF01217690
S. J. Li and C. R. Chen, “Higher order optimality conditions for Henig efficient solutions in set-valued optimization,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1184–1200, 2006. doi: 10.1016/j.jmaa.2005.11.035
J. Borwein, “Multivalued convexity and optimization: a unified approach to inequality and equality constraints,” Mathematical Programming, vol. 13, no. 1, pp. 183–199, 1977. doi: 10.1007/BF01584336
P. Q. Khanh and N. M. Tung, “Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints,” Optimization, vol. 64, no. 7, pp. 1547–1575, 2014. doi: 10.1080/02331934.2014.886036
K. Das, S. Treanta, and M. B. Khan, “Set-valued fractional programming problems with σ-arcwisely connectivity,” AIMS Mathematics, vol. 8, no. 6, pp. 13181–13204, 2023. doi: 10.3934/math.2023666
H. W. Corley, “Existence and Lagrangian duality for maximizations of set-valued functions,” Journal of Optimization Theory and Applications, vol. 54, no. 3, pp. 489–501, 1987. doi: 10.1007/BF00940198
K. Das, “Sufficiency and duality of set-valued fractional programming problems via secondorder contingent epiderivative,” Yugoslav Journal of Operations Research, vol. 32, no. 2, pp. 167–188, 2022. doi: 10.2298/YJOR210218019D
K. Das and C. Nahak, “Set-valued optimization problems via second-order contingent epiderivative,” Yugoslav Journal of Operations Research, vol. 31, no. 1, pp. 75–94, 2021. doi: 10.2298/YJOR191215041D
N. Pokharna and I. P. Tripathi, “Optimality and duality for E-minimax fractional programming: application to multiobjective optimization,” Journal of Applied Mathematics and Computing, vol. 69, no. 3, pp. 2361–2388, 2023. doi: 10.1007/s12190-023-01838-y
K. Das and C. Nahak, “Optimality conditions for set-valued minimax fractional programming problems,” SeMAJournal, vol. 77, no. 2, pp. 161–179, 2020. doi: 10.1007/s40324-019-002097
K. Das, S. Treanta, and T. Saeed, “Mond-Weir and Wolfe duality of set-valued fractional minimax problems in terms of contingent epi-derivative of second-order,” Mathematics, vol. 10, no. 6:938, pp. 1–21, 2022. doi: 10.3390/math10060938
K. Das, “Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness,” Control & Cybernetics, vol. 51, no. 1, pp. 43–69, 2022. doi: 10.2478/candc2022-0004
S. R. Yadav and R. N. Mukherjee, “Duality for fractional minimax programming problems,” Journal of the Australian Mathematical Society (Series B), vol. 31, no. 4, pp. 484–492, 1990. doi: 10.1017/S0334270000006809
S. Chandra and V. Kumar, “Duality in fractional minimax programming,” Journal of the Australian Mathematical Society (Series A), vol. 58, no. 3, pp. 376–386, 1995. doi: 10.1017/S1446788700038362
T. Weir, “Pseudoconvex minimax programming,” Utilitas Mathematica, vol. 42, pp. 234–240, 1992.
C. R. Bector and B. L. Bhatia, “Sufficient optimality conditions and duality for a minmax problem,” Utilitas Mathematica, vol. 27, pp. 229–247, 1985.
G. J. Zalmai, “Optimality criteria and duality for a class of minimax programming problems with generalized invexity conditions,” Utilitas Mathematica, vol. 32, pp. 35–57, 1987.
J. C. Liu and C. S. Wu, “On minimax fractional optimality conditions with (F,ρ)-convexity,” Journal of Mathematical Analysis and Applications, vol. 219, no. 1, pp. 36–51, 1998. doi: 10.1006/jmaa.1997.5785
I. Ahmad, “Optimality conditions and duality in fractional minimax programming involving generalized ρ-invexity,” Journal of Management Information Systems, vol. 19, pp. 165–180, 2003.
Z. A. Liang and Z. W. Shi, “Optimality conditions and duality for minimax fractional programming with generalized convexity,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 474–488, 2003. doi: 10.1016/S0022-247X(02)00553-X
H. C. Lai, J. C. Liu, and K. Tanaka, “Necessary and sufficient conditions for minimax fractional programming,” Journal of Mathematical Analysis and Applications, vol. 230, no. 2, pp. 311–328, 1999. doi: 10.1006/jmaa.1998.6204
H. C. Lai and J. C. Lee, “On duality theorems for nondifferentiable minimax fractional programming,” Journal of Computational and Applied Mathematics, vol. 146, no. 1, pp. 115–126, 2002. doi: 10.1016/S0377-0427(02)00422-3
I. Ahmad and Z. Husain, “Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity,” Journal of Optimization Theory and Applications, vol. 129, no. 2, pp. 255–275, 2006. doi: 10.1007/s10957-006-9057-0
S. J. Li, K. L. Teo, and X. Q. Yang, “Higher-order Mond-Weir duality for set-valued optimization,” Journal of Computational and Applied Mathematics, vol. 217, no. 2, pp. 339–349, 2008. doi: 10.1016/j.cam.2007.02.011
S. J. Li, K. L. Teo, and X. Q. Yang,, “Higher-order optimality conditions for set-valued optimization,” Journal of Optimization Theory and Applications, vol. 137, no. 3, pp. 533–553, 2008. doi: 10.1007/s10957-0079345-3
M. Avriel, Nonlinear Programming: Theory and Method. Englewood Cliffs, New Jersey: Prentice-Hall, 1976.
J. Y. Fu and Y. H. Wang, “Arcwise connected cone-convex functions and mathematical programming,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 339–352, 2003. doi: 10.1023/A:1025451422581
C. S. Lalitha, J. Dutta, and M. G. Govil, “Optimality criteria in set-valued optimization,” Journal of the Australian Mathematical Society, vol. 75, no. 2, pp. 221–232, 2003. doi: 10.1017/S1446788700003736
K.DasandC.Nahak,“Sufficiencyanddualityofset-valued optimization problems via higherorder contingent derivative,” Journal of Advanced Mathematical Studies, vol. 8, no. 1, pp. 137–151, 2015.
J. P. Aubin, Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions. Paris: Universit´e Paris IX-Dauphine, Centre de recherche de math´ematiques de la d´ecision, 1980.
J. P. Aubin and H. Frankowska, Set-Valued Analysis. Boston: Birh¨auser, 1990.
J. Jahn and R. Rauh, “Contingent epiderivatives and set-valued optimization,” Mathematical Methods of Operations Research, vol. 46, no. 2, pp. 193–211, 1997. doi: 10.1007/BF01217690
S. J. Li and C. R. Chen, “Higher order optimality conditions for Henig efficient solutions in set-valued optimization,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1184–1200, 2006. doi: 10.1016/j.jmaa.2005.11.035
J. Borwein, “Multivalued convexity and optimization: a unified approach to inequality and equality constraints,” Mathematical Programming, vol. 13, no. 1, pp. 183–199, 1977. doi: 10.1007/BF01584336
P. Q. Khanh and N. M. Tung, “Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints,” Optimization, vol. 64, no. 7, pp. 1547–1575, 2014. doi: 10.1080/02331934.2014.886036
K. Das, S. Treanta, and M. B. Khan, “Set-valued fractional programming problems with σ-arcwisely connectivity,” AIMS Mathematics, vol. 8, no. 6, pp. 13181–13204, 2023. doi: 10.3934/math.2023666
H. W. Corley, “Existence and Lagrangian duality for maximizations of set-valued functions,” Journal of Optimization Theory and Applications, vol. 54, no. 3, pp. 489–501, 1987. doi: 10.1007/BF00940198
K. Das, “Sufficiency and duality of set-valued fractional programming problems via secondorder contingent epiderivative,” Yugoslav Journal of Operations Research, vol. 32, no. 2, pp. 167–188, 2022. doi: 10.2298/YJOR210218019D
K. Das and C. Nahak, “Set-valued optimization problems via second-order contingent epiderivative,” Yugoslav Journal of Operations Research, vol. 31, no. 1, pp. 75–94, 2021. doi: 10.2298/YJOR191215041D
N. Pokharna and I. P. Tripathi, “Optimality and duality for E-minimax fractional programming: application to multiobjective optimization,” Journal of Applied Mathematics and Computing, vol. 69, no. 3, pp. 2361–2388, 2023. doi: 10.1007/s12190-023-01838-y
K. Das and C. Nahak, “Optimality conditions for set-valued minimax fractional programming problems,” SeMAJournal, vol. 77, no. 2, pp. 161–179, 2020. doi: 10.1007/s40324-019-002097
K. Das, S. Treanta, and T. Saeed, “Mond-Weir and Wolfe duality of set-valued fractional minimax problems in terms of contingent epi-derivative of second-order,” Mathematics, vol. 10, no. 6:938, pp. 1–21, 2022. doi: 10.3390/math10060938
K. Das, “Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness,” Control & Cybernetics, vol. 51, no. 1, pp. 43–69, 2022. doi: 10.2478/candc2022-0004
Published
2024-10-07
How to Cite
DAS, Koushik.
Higher-Order Mond-Weir Duality of Set-Valued Fractional Minimax Programming Problems.
Yugoslav Journal of Operations Research, [S.l.], v. 35, n. 4, p. 749-774, oct. 2024.
ISSN 2334-6043.
Available at: <https://yujor.fon.bg.ac.rs/index.php/yujor/article/view/1302>. Date accessed: 22 nov. 2025.
doi: https://doi.org/10.2298/YJOR231215046D.
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