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March
2022, 12(1): 121-134.
doi: 10.3934/naco.2021055
Optimality and duality for complex multi-objective programming
| 1. | No. 100, Wenhwa Road, Seatweeen, Department of Applied Mathematics, Feng Chia University, Taichung, 40724, Taiwan |
| 2. | Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan |
We consider a complex multi-objective programming problem (CMP). In order to establish the optimality conditions of problem (CMP), we introduce several properties of optimal efficient solutions and scalarization techniques. Furthermore, a certain parametric dual model is discussed, and their duality theorems are proved.
Mathematics Subject Classification:
Primary: 90C46, 90C29; Secondary: 49K35.
Citation:
Tone-Yau Huang, Tamaki Tanaka. Optimality and duality for complex multi-objective programming. Numerical Algebra, Control and Optimization,
2022, 12
(1)
: 121-134.
doi: 10.3934/naco.2021055
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References:
| [1] |
R. A. Abrams,
Nonlinear programming in complex space: sufficient conditions and duality, J. Math. Anal. Appl., 38 (1972), 619-632.
doi: 10.1016/0022-247X(72)90073-X. |
| [2] |
R. A. Abrams and A. Ben-Israel, Complex mathematical programming, Developments in Operations Research (eds. B. Avi-Itzhak, Gordon and Breach), New York, (1971), 3–20. |
| [3] |
R. A. Abrams and A. Ben-Israel,
Nonlinear programming in complex space: necessary conditions, SIAM J. Control., 9 (1971), 606-620.
|
| [4] |
N. Datta and D. Bhatia,
Duality for a class of nondifferentiable mathematical programming problems in complex space, J. Math. Anal. Appl., 101 (1984), 1-11.
doi: 10.1016/0022-247X(84)90053-2. |
| [5] |
D. I. Duca, On vectorial programming problem in complex space, Studia Univ. Babeș-Bolyai Math., 24 (1979), 51–56. |
| [6] |
D. I. Duca, Proper efficiency in the complex vectorial programming, Studia Univ. Babeș-Bolyai Math., 25 (1980), 73–80. |
| [7] |
D. I. Duca, Efficiency criteria in vectorial programming in complex space without convexity, Cahiers Centre Études Rech. Opér., 26 (1984), 217–226. |
| [8] |
D. I. Duca, Multicriteria Optimization in Complex Space, Casa Cǎrţii de Ştiinţǎ, Cluj-Napoca, 2005. |
| [9] |
M. E. Elbrolosy,
Efficiency for a generalized form of vector optimization problems in complex space, Optimization, 65 (2016), 1245-1257.
doi: 10.1080/02331934.2015.1104680. |
| [10] |
O. Ferrero,
On nonlinear programming in complex spaces, J. Math. Anal. Appl., 164 (1992), 399-416.
doi: 10.1016/0022-247X(92)90123-U. |
| [11] |
H. C. Lai and T. Y. Huang,
Optimality conditions for a nondifferentiable minimax programming in complex spaces, Nonlinear Anal., 71 (2009), 1205-1212.
doi: 10.1016/j.na.2008.11.053. |
| [12] |
H. C. Lai and T. Y. Huang,
Optimality conditions for nondifferentiable minimax fractional programming with complex variables, J. Math. Anal. Appl., 359 (2009), 229-239.
doi: 10.1016/j.jmaa.2009.05.049. |
| [13] |
H. C. Lai and T. Y. Huang,
Nondifferentiable minimax fractional programming in complex spaces with parametric duality, J. Global Optim., 53 (2012), 243-254.
doi: 10.1007/s10898-011-9680-7. |
| [14] |
H. C. Lai and T. Y. Huang,
Mixed type duality for a nondifferentiable minimax fractional complex programming, Pacific J. Optim., 10 (2014), 305-319.
|
| [15] |
H. C. Lai and J. C. Liu, Duality for nondifferentiable minimax programming in complex spaces, Nonlinear Anal., 71 (2009), e224–e233.
doi: 10.1016/j.na.2008.10.062. |
| [16] |
N. Levinson,
Linear programming in complex space, J. Math. Anal. Appl., 14 (1966), 44-62.
doi: 10.1016/0022-247X(66)90061-8. |
| [17] |
B. Mond and B. D. Craven,
A class of nondifferentiable complex programming problems, J. Math. Oper. and Stat., 6 (1975), 581-591.
doi: 10.1007/bf01966096. |
| [18] |
I. M. Stancu-Minasian, D. I. Duca and T. Nishida,
Multiple objective linear fractional optimization in complex space, Math. Japonica., 35 (1990), 195-203.
|
| [19] |
Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, Orlando, FL, 1985.
|
| [20] |
E. A. Youness and M. E. Elbrolosy,
Extension to necessary optimality conditions in complex programming, Appl. Math. Comput., 154 (2004), 229-237.
doi: 10.1016/S0096-3003(03)00706-9. |
| [21] |
E. A. Youness and M. E. Elbrolosy,
Extension to sufficient optimality conditions in complex programming, J. Math. Stat., 1 (2005), 40-48.
doi: 10.3844/jmssp.2005.40.48. |
show all references
References:
| [1] |
R. A. Abrams,
Nonlinear programming in complex space: sufficient conditions and duality, J. Math. Anal. Appl., 38 (1972), 619-632.
doi: 10.1016/0022-247X(72)90073-X. |
| [2] |
R. A. Abrams and A. Ben-Israel, Complex mathematical programming, Developments in Operations Research (eds. B. Avi-Itzhak, Gordon and Breach), New York, (1971), 3–20. |
| [3] |
R. A. Abrams and A. Ben-Israel,
Nonlinear programming in complex space: necessary conditions, SIAM J. Control., 9 (1971), 606-620.
|
| [4] |
N. Datta and D. Bhatia,
Duality for a class of nondifferentiable mathematical programming problems in complex space, J. Math. Anal. Appl., 101 (1984), 1-11.
doi: 10.1016/0022-247X(84)90053-2. |
| [5] |
D. I. Duca, On vectorial programming problem in complex space, Studia Univ. Babeș-Bolyai Math., 24 (1979), 51–56. |
| [6] |
D. I. Duca, Proper efficiency in the complex vectorial programming, Studia Univ. Babeș-Bolyai Math., 25 (1980), 73–80. |
| [7] |
D. I. Duca, Efficiency criteria in vectorial programming in complex space without convexity, Cahiers Centre Études Rech. Opér., 26 (1984), 217–226. |
| [8] |
D. I. Duca, Multicriteria Optimization in Complex Space, Casa Cǎrţii de Ştiinţǎ, Cluj-Napoca, 2005. |
| [9] |
M. E. Elbrolosy,
Efficiency for a generalized form of vector optimization problems in complex space, Optimization, 65 (2016), 1245-1257.
doi: 10.1080/02331934.2015.1104680. |
| [10] |
O. Ferrero,
On nonlinear programming in complex spaces, J. Math. Anal. Appl., 164 (1992), 399-416.
doi: 10.1016/0022-247X(92)90123-U. |
| [11] |
H. C. Lai and T. Y. Huang,
Optimality conditions for a nondifferentiable minimax programming in complex spaces, Nonlinear Anal., 71 (2009), 1205-1212.
doi: 10.1016/j.na.2008.11.053. |
| [12] |
H. C. Lai and T. Y. Huang,
Optimality conditions for nondifferentiable minimax fractional programming with complex variables, J. Math. Anal. Appl., 359 (2009), 229-239.
doi: 10.1016/j.jmaa.2009.05.049. |
| [13] |
H. C. Lai and T. Y. Huang,
Nondifferentiable minimax fractional programming in complex spaces with parametric duality, J. Global Optim., 53 (2012), 243-254.
doi: 10.1007/s10898-011-9680-7. |
| [14] |
H. C. Lai and T. Y. Huang,
Mixed type duality for a nondifferentiable minimax fractional complex programming, Pacific J. Optim., 10 (2014), 305-319.
|
| [15] |
H. C. Lai and J. C. Liu, Duality for nondifferentiable minimax programming in complex spaces, Nonlinear Anal., 71 (2009), e224–e233.
doi: 10.1016/j.na.2008.10.062. |
| [16] |
N. Levinson,
Linear programming in complex space, J. Math. Anal. Appl., 14 (1966), 44-62.
doi: 10.1016/0022-247X(66)90061-8. |
| [17] |
B. Mond and B. D. Craven,
A class of nondifferentiable complex programming problems, J. Math. Oper. and Stat., 6 (1975), 581-591.
doi: 10.1007/bf01966096. |
| [18] |
I. M. Stancu-Minasian, D. I. Duca and T. Nishida,
Multiple objective linear fractional optimization in complex space, Math. Japonica., 35 (1990), 195-203.
|
| [19] |
Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, Orlando, FL, 1985.
|
| [20] |
E. A. Youness and M. E. Elbrolosy,
Extension to necessary optimality conditions in complex programming, Appl. Math. Comput., 154 (2004), 229-237.
doi: 10.1016/S0096-3003(03)00706-9. |
| [21] |
E. A. Youness and M. E. Elbrolosy,
Extension to sufficient optimality conditions in complex programming, J. Math. Stat., 1 (2005), 40-48.
doi: 10.3844/jmssp.2005.40.48. |
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