-
Previous Article
The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications
- JIMO Home
- This Issue
-
Next Article
CVaR-based robust models for portfolio selection
Higher-order symmetric duality for multiobjective programming with cone constraints
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[1] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[2] |
Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 |
[3] |
Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 |
[4] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
[5] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[6] |
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 |
[7] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020378 |
[8] |
A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 |
[9] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]