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July  2020, 16(4): 1873-1884. doi: 10.3934/jimo.2019033

Higher-order symmetric duality for multiobjective programming with cone constraints

Liping Tang 1,2,3, , Xinmin Yang 1, and  Ying Gao 1,

1. 

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
2. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
3. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received  April 2018 Revised  August 2018 Published  July 2020 Early access  May 2019

In this work, a pair of higher-order symmetric dual multiobjective optimization problems is formulated. Weak, strong and converse duality theorems are established under suitable assumptions. Some examples are also given to illustrate our main results. Furthermore, certain deficiencies in the formulations and the proof of the work of Kassem [Applied Mathematics and Computation, 209 (2009), 405-409] are pointed out.

Keywords: Multiobjective programming, higher-order symmetric duality, duality theorem.
Mathematics Subject Classification: Primary: 90C26, 90C29, 90C30; Secondary: 90C46.
Citation: Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1873-1884. doi: 10.3934/jimo.2019033
References:
[1]

R. P. Agarwal, I. Ahmad and S. K. Gupta, A note on higher-order nondifferentiable symmetric duality in multiobjective programming, Applied Mathematics Letters, 24 (2011), 1308-1311.  doi: 10.1016/j.aml.2011.02.021.  Google Scholar

[2]

I. Ahmad, Unified higher order duality in nondifferentiable multiobjective programming involving cones, Mathematical and Computer Modelling, 55 (2012), 419-425.  doi: 10.1016/j.mcm.2011.08.020.  Google Scholar

[3]

T. Antczak and G. J. Zalmai, Second order (Φ, ρ)-V-invexity and duality for semi-infinite minimax fractional programming, Applied Mathematics and Computation, 227 (2014), 831-856.  doi: 10.1016/j.amc.2013.10.050.  Google Scholar

[4]

M. S. Bazaraa and J. J. Goode, On symmetric duality in nonlinear programming, Operations Research, 21 (1973), 1-9.  doi: 10.1287/opre.21.1.1.  Google Scholar

[5]

S. Chandra and V. Kumar, A note on pseudo-invexity and symmetric duality, European Journal of Operational Research, 105 (1998), 626-629.   Google Scholar

[6]

G. B. Dantzig, E. Eisenberg and R. W. Cottle, Symmetric dual non-linear programs, Pacific Journal of Mathematics, 23 (1965), 265-269.  doi: 10.2140/pjm.1965.15.809.  Google Scholar

[7]

I. P. Debnath, S. K. Gupta and I. Ahmad, A note on strong duality theorem for a multiobjective higher order nondifferentiable symmetric dual programs, Opsearch, 53 (2016), 151-156.  doi: 10.1007/s12597-015-0221-x.  Google Scholar

[8]

W. S. Dorn, A symmetric dual theorem for quadratic programming, Journal of the Operations Research Society of Japan, 2 (1960), 93-97.   Google Scholar

[9]

Y. Gao, Higher-order symmetric duality in multiobjective programming problems, Acta Mathematicae Applicatae Sinica, English Series, 32 (2016), 485-494.  doi: 10.1007/s10255-016-0578-5.  Google Scholar

[10]

S. K. Gupta and A. Jayswal, Multiobjective higher-order symmetric duality involving generalized cone-invex functions, Computers & Mathematics with Applications, 60 (2010), 3187-3192.  doi: 10.1016/j.camwa.2010.10.023.  Google Scholar

[11]

A. Jayswal, I. Ahmad and A. K. Prasad, Higher Order Fractional Symmetric Duality Over Cone Constraints, Journal of Mathematical Modelling and Algorithms in Operations Research, 14 (2015), 91-101.  doi: 10.1007/s10852-014-9259-7.  Google Scholar

[12]

M. A. E. H. Kassem, Higher-order symmetric duality in vector optimization problem involving generalized cone-invex functions, Applied Mathematics and Computation, 209 (2009), 405-409.  doi: 10.1016/j.amc.2008.12.063.  Google Scholar

[13]

O. L. Mangasarian, Second and higher order duality in nonlinear programming problem, Journal of Mathematical Analysis and Applications, 51 (1975), 607-620.  doi: 10.1016/0022-247X(75)90111-0.  Google Scholar

[14]

S. K. Mishra and K. K. Lai, Second order symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research, 178 (2007), 20-26.  doi: 10.1016/j.ejor.2005.11.024.  Google Scholar

[15]

B. Mond and T. Weir, Generalized concavity and duality, in Generalized Concavity in Optimization and Economics (eds. S. Schaible and W. T. Ziemba), Academic Press, (1981), 263–279. Google Scholar

[16]

S. K. Padhan and C. Nahak, Higher-order symmetric duality in multiobjective programming problems under higher-order invexity, Applied Mathematics and Computation, 218 (2011), 1705-1712.  doi: 10.1016/j.amc.2011.06.049.  Google Scholar

[17]

S. K. Suneja, S. Aggarwal and S. Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research, 141 (2002), 471-479.  doi: 10.1016/S0377-2217(01)00258-2.  Google Scholar

[18]

S. K. Suneja and P. Louhan, Higher-order symmetric duality under cone-invexity and other related concepts, Journal of Computational and Applied Mathematics, 255 (2014), 825-836.  doi: 10.1016/j.cam.2013.07.003.  Google Scholar

[19]

L. P. Tang, H. Yan and X. M. Yang, Second order duality for multiobjective programming with cone constraints, Science China Mathematics, 59 (2016), 1285-1306.  doi: 10.1007/s11425-016-5147-0.  Google Scholar

[20]

X. M. Yang, X. Q. Yang and K. L. Teo, Higher-order symmetric duality in multiobjective mathematical programming with invexity, Journal of Industrial and Management Optimization, 4 (2008), 335-391.  doi: 10.3934/jimo.2008.4.385.  Google Scholar

[21]

X. M. Yang, J. Yang, T. L. Yip and K. L. Teo, Higher-order Mond-Weir converse duality in multiobjective programming involving cones, Science China Mathematics, 56 (2013), 2389-2392.  doi: 10.1007/s11425-013-4700-3.  Google Scholar

[22]

X. M. Yang, J. Yang and H. W. J. Lee, Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs, Journal of Industrial and Management Optimization, 9 (2013), 525-530.  doi: 10.3934/jimo.2013.9.525.  Google Scholar

[23]

X. M. Yang and K. L. Teo, A converse duality theorem on higher-order dual models in nondifferentiable mathematical programming, Optimization Letters, 6 (2012), 11-15.  doi: 10.1007/s11590-010-0247-1.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, I. Ahmad and S. K. Gupta, A note on higher-order nondifferentiable symmetric duality in multiobjective programming, Applied Mathematics Letters, 24 (2011), 1308-1311.  doi: 10.1016/j.aml.2011.02.021.  Google Scholar

[2]

I. Ahmad, Unified higher order duality in nondifferentiable multiobjective programming involving cones, Mathematical and Computer Modelling, 55 (2012), 419-425.  doi: 10.1016/j.mcm.2011.08.020.  Google Scholar

[3]

T. Antczak and G. J. Zalmai, Second order (Φ, ρ)-V-invexity and duality for semi-infinite minimax fractional programming, Applied Mathematics and Computation, 227 (2014), 831-856.  doi: 10.1016/j.amc.2013.10.050.  Google Scholar

[4]

M. S. Bazaraa and J. J. Goode, On symmetric duality in nonlinear programming, Operations Research, 21 (1973), 1-9.  doi: 10.1287/opre.21.1.1.  Google Scholar

[5]

S. Chandra and V. Kumar, A note on pseudo-invexity and symmetric duality, European Journal of Operational Research, 105 (1998), 626-629.   Google Scholar

[6]

G. B. Dantzig, E. Eisenberg and R. W. Cottle, Symmetric dual non-linear programs, Pacific Journal of Mathematics, 23 (1965), 265-269.  doi: 10.2140/pjm.1965.15.809.  Google Scholar

[7]

I. P. Debnath, S. K. Gupta and I. Ahmad, A note on strong duality theorem for a multiobjective higher order nondifferentiable symmetric dual programs, Opsearch, 53 (2016), 151-156.  doi: 10.1007/s12597-015-0221-x.  Google Scholar

[8]

W. S. Dorn, A symmetric dual theorem for quadratic programming, Journal of the Operations Research Society of Japan, 2 (1960), 93-97.   Google Scholar

[9]

Y. Gao, Higher-order symmetric duality in multiobjective programming problems, Acta Mathematicae Applicatae Sinica, English Series, 32 (2016), 485-494.  doi: 10.1007/s10255-016-0578-5.  Google Scholar

[10]

S. K. Gupta and A. Jayswal, Multiobjective higher-order symmetric duality involving generalized cone-invex functions, Computers & Mathematics with Applications, 60 (2010), 3187-3192.  doi: 10.1016/j.camwa.2010.10.023.  Google Scholar

[11]

A. Jayswal, I. Ahmad and A. K. Prasad, Higher Order Fractional Symmetric Duality Over Cone Constraints, Journal of Mathematical Modelling and Algorithms in Operations Research, 14 (2015), 91-101.  doi: 10.1007/s10852-014-9259-7.  Google Scholar

[12]

M. A. E. H. Kassem, Higher-order symmetric duality in vector optimization problem involving generalized cone-invex functions, Applied Mathematics and Computation, 209 (2009), 405-409.  doi: 10.1016/j.amc.2008.12.063.  Google Scholar

[13]

O. L. Mangasarian, Second and higher order duality in nonlinear programming problem, Journal of Mathematical Analysis and Applications, 51 (1975), 607-620.  doi: 10.1016/0022-247X(75)90111-0.  Google Scholar

[14]

S. K. Mishra and K. K. Lai, Second order symmetric duality in multiobjective programming involving generalized cone-invex functions, European Journal of Operational Research, 178 (2007), 20-26.  doi: 10.1016/j.ejor.2005.11.024.  Google Scholar

[15]

B. Mond and T. Weir, Generalized concavity and duality, in Generalized Concavity in Optimization and Economics (eds. S. Schaible and W. T. Ziemba), Academic Press, (1981), 263–279. Google Scholar

[16]

S. K. Padhan and C. Nahak, Higher-order symmetric duality in multiobjective programming problems under higher-order invexity, Applied Mathematics and Computation, 218 (2011), 1705-1712.  doi: 10.1016/j.amc.2011.06.049.  Google Scholar

[17]

S. K. Suneja, S. Aggarwal and S. Davar, Multiobjective symmetric duality involving cones, European Journal of Operational Research, 141 (2002), 471-479.  doi: 10.1016/S0377-2217(01)00258-2.  Google Scholar

[18]

S. K. Suneja and P. Louhan, Higher-order symmetric duality under cone-invexity and other related concepts, Journal of Computational and Applied Mathematics, 255 (2014), 825-836.  doi: 10.1016/j.cam.2013.07.003.  Google Scholar

[19]

L. P. Tang, H. Yan and X. M. Yang, Second order duality for multiobjective programming with cone constraints, Science China Mathematics, 59 (2016), 1285-1306.  doi: 10.1007/s11425-016-5147-0.  Google Scholar

[20]

X. M. Yang, X. Q. Yang and K. L. Teo, Higher-order symmetric duality in multiobjective mathematical programming with invexity, Journal of Industrial and Management Optimization, 4 (2008), 335-391.  doi: 10.3934/jimo.2008.4.385.  Google Scholar

[21]

X. M. Yang, J. Yang, T. L. Yip and K. L. Teo, Higher-order Mond-Weir converse duality in multiobjective programming involving cones, Science China Mathematics, 56 (2013), 2389-2392.  doi: 10.1007/s11425-013-4700-3.  Google Scholar

[22]

X. M. Yang, J. Yang and H. W. J. Lee, Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs, Journal of Industrial and Management Optimization, 9 (2013), 525-530.  doi: 10.3934/jimo.2013.9.525.  Google Scholar

[23]

X. M. Yang and K. L. Teo, A converse duality theorem on higher-order dual models in nondifferentiable mathematical programming, Optimization Letters, 6 (2012), 11-15.  doi: 10.1007/s11590-010-0247-1.  Google Scholar

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