American Institute of Mathematical Sciences

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March  2022, 12(1): 121-134. doi: 10.3934/naco.2021055

Optimality and duality for complex multi-objective programming

Tone-Yau Huang 1,, and  Tamaki Tanaka 2,

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.

Keywords: Multi-objective programming, efficient solutions, generalized convexity, duality problem.
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 & 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.  Google Scholar

[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. Google Scholar

[3]

R. A. Abrams and A. Ben-Israel, Nonlinear programming in complex space: necessary conditions, SIAM J. Control., 9 (1971), 606-620.   Google Scholar

[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.  Google Scholar

[5]

D. I. Duca, On vectorial programming problem in complex space, Studia Univ. Babeș-Bolyai Math., 24 (1979), 51–56.  Google Scholar

[6]

D. I. Duca, Proper efficiency in the complex vectorial programming, Studia Univ. Babeș-Bolyai Math., 25 (1980), 73–80.  Google Scholar

[7]

D. I. Duca, Efficiency criteria in vectorial programming in complex space without convexity, Cahiers Centre Études Rech. Opér., 26 (1984), 217–226.  Google Scholar

[8]

D. I. Duca, Multicriteria Optimization in Complex Space, Casa Cǎrţii de Ştiinţǎ, Cluj-Napoca, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

N. Levinson, Linear programming in complex space, J. Math. Anal. Appl., 14 (1966), 44-62.  doi: 10.1016/0022-247X(66)90061-8.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, Orlando, FL, 1985.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[3]

R. A. Abrams and A. Ben-Israel, Nonlinear programming in complex space: necessary conditions, SIAM J. Control., 9 (1971), 606-620.   Google Scholar

[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.  Google Scholar

[5]

D. I. Duca, On vectorial programming problem in complex space, Studia Univ. Babeș-Bolyai Math., 24 (1979), 51–56.  Google Scholar

[6]

D. I. Duca, Proper efficiency in the complex vectorial programming, Studia Univ. Babeș-Bolyai Math., 25 (1980), 73–80.  Google Scholar

[7]

D. I. Duca, Efficiency criteria in vectorial programming in complex space without convexity, Cahiers Centre Études Rech. Opér., 26 (1984), 217–226.  Google Scholar

[8]

D. I. Duca, Multicriteria Optimization in Complex Space, Casa Cǎrţii de Ştiinţǎ, Cluj-Napoca, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

N. Levinson, Linear programming in complex space, J. Math. Anal. Appl., 14 (1966), 44-62.  doi: 10.1016/0022-247X(66)90061-8.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multiobjective Optimization, Academic Press, Orlando, FL, 1985.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The graphs of Example 2.1
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