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

* Corresponding author: Tone-Yau Huang

Received  March 2020 Revised  July 2021 Published  March 2022 Early access  November 2021

Fund Project: The first author is supported by MOST 109-2115-M-035-002, Taiwan

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.

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
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-MinasianD. I. Duca and T. Nishida, Multiple objective linear fractional optimization in complex space, Math. Japonica., 35 (1990), 195-203. 

[19] Y. SawaragiH. 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-MinasianD. I. Duca and T. Nishida, Multiple objective linear fractional optimization in complex space, Math. Japonica., 35 (1990), 195-203. 

[19] Y. SawaragiH. 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.

Figure 1.  The graphs of Example 2.1
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