American Institute of Mathematical Sciences

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September  2021, 17(5): 2669-2683. doi: 10.3934/jimo.2020088

A combined scalarization method for multi-objective optimization problems

Yuan-mei Xia 1, , Xin-min Yang 2,, and  Ke-quan Zhao 2,

1. 

Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China
2. 

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

* Corresponding author: Xin-min Yang

Received  October 2019 Revised  January 2020 Published  September 2021 Early access  April 2020

Fund Project: This research was partially supported by the major program of National Natural Science Foundation of China (No.11991024), the National Natural Science Foundation of China (Nos. 11971084, 11671062, 11771064), the Science Foundation of Chongqing (Nos. cstc2015jcyjA00027, cstc2017jcyj-yszxX0008, cstc2018jcyj-yszx0007) and the Open Project Funded by the Key Lab for OCME, School of Mathematical Sciences, Chongqing Normal University (CSSXKFKTQ201810)

In this paper, we propose a new combined scalarization method of multi-objective optimization problems by using the surplus variables and the generalized Tchebycheff norm and then use it to obtain some equivalent scalarization characterizations of (weakly, strictly, properly) efficient solutions by adjusting the range of parameters. These scalarization results do not need any convexity assumption conditions of objective functions. Furthermore, we establish some scalarization results of approximate solutions by means of the method. Moreover, we also present some examples to illustrate the main results.

Keywords: Multi-objective optimization, Tchebycheff norm, surplus variables, combined scalarization, (weakly, properly, approximate) efficient solution.
Mathematics Subject Classification: 90C26, 90C29, 90C30, 90C46.
Citation: Yuan-mei Xia, Xin-min Yang, Ke-quan Zhao. A combined scalarization method for multi-objective optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2669-2683. doi: 10.3934/jimo.2020088
References:
[1]

F. Akbari, M. Ghaznavi and E. Khorram, A revised Pascoletti-Serafini scalarization method for multiobjective optimization problems, Journal of Optimization Theory and Applications, 178 (2018), 560-590.  doi: 10.1007/s10957-018-1289-2.

[2]

H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 71 (1979), 232-241.  doi: 10.1016/0022-247X(79)90226-9.

[3]

R. S. Burachik, C. Y. Kaya and M. M. Rizvi, A new scalarization technique to approximate Pareto fronts of problems with disconnected feasible sets, Journal of Optimization Theory and Applications, 162 (2014), 428-446.  doi: 10.1007/s10957-013-0346-0.

[4]

G. Y. Chen, Necessary conditions of nondominated solutions in multicriteria decision making, Journal of Mathematical Analysis and Applications, 104 (1984), 38-46.  doi: 10.1016/0022-247X(84)90027-1.

[5]

E. U. Choo and D. R. Atkins, Proper efficiency in nonconvex multicriteria programming, Mathematics of Operations Research, 8 (1983), 467-470. 

[6]

A. Chinchuluun and P. M. Pardalos, A survey of recent developments in multiobjective optimization, Annals of Operations Research, 154 (2007), 29-50.  doi: 10.1007/s10479-007-0186-0.

[7]

Y. Collette and P. Siarry, Multiobjective Optimization. Principles and Case Studies, , Springer, Berlin, 2003.

[8]

J. Dutta and C. Y. Kaya, A new scalarization and numerical method for constructing the weak pareto front of multi-objective optimization problems, Optimization, 60 (2011), 1091-1104.  doi: 10.1080/02331934.2011.587006.

[9]

M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.

[10]

M. Ehrgott and S. Ruzika, Improved $\varepsilon$-constraint method for multiobjective programming, Journal of Optimization Theory and Applications, 138 (2008), 375-396.  doi: 10.1007/s10957-008-9394-2.

[11]

A. Engau and M. M. Wiecek, Generating $\epsilon$-efficient solutions in multiobjective programming, European Journal of Operational Research, 177 (2007), 1566-1579.  doi: 10.1016/j.ejor.2005.10.023.

[12]

A. M. Geoffrion, Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications, 22 (1968), 618-630.  doi: 10.1016/0022-247X(68)90201-1.

[13]

A. Ghane-Kanafi and E. Khorram, A new scalarization method for finding the efficient frontier in non-convex multi-objective problems, Applied Mathematical Modelling, 39 (2015), 7483-7498.  doi: 10.1016/j.apm.2015.03.022.

[14]

B. A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization, Optimization, 62 (2013), 703-720.  doi: 10.1080/02331934.2012.668190.

[15]

S. M. Guu, N. J. Huang and J. Li, Scalarization approaches for set-valued vector optimization problems and vector variational inequalities, Journal of Mathematical Analysis and Applications, 356 (2009), 564-576.  doi: 10.1016/j.jmaa.2009.03.040.

[16]

X. X. Huang and X. Q. Yang, On characterizations of proper efficiency for nonconvex multiobjective optimization, Journal of Global Optimization, 23 (2002), 213-231.  doi: 10.1023/A:1016522528364.

[17]

R. Kasimbeyli, Z. K. Ozturk, N. Kasimbeyli and et al., Comparison of some scalarization methods in multiobjective optimization, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1875-1905.  doi: 10.1007/s40840-017-0579-4.

[18]

S. S. Kutateladze, Convex $\varepsilon$-programming, Dokl. Akad. Nauk SSSR, 245 (1979), 1048-1050. 

[19]

P. Loridan, $\epsilon$-solutions in vector minimization problems, Journal of Optimization Theory and Applications, 43 (1984), 265-276.  doi: 10.1007/BF00936165.

[20]

D. T. Luc, T. Q. Phong and M. Volle, Scalarizing functions for generating the weakly efficient solution set in convex multiobjective problems, SIAM Journal on Optimization, 15 (2005), 987-1001.  doi: 10.1137/040603097.

[21]

S. Mishra and M. Noor, Some nondifferentiable multiobjective programming problems, Journal of Mathematical Analysis and Applications, 316 (2006), 472-482.  doi: 10.1016/j.jmaa.2005.04.067.

[22]

N. Rastegar and E. Khorram, A combined scalarizing method for multiobjective programming problems, European Journal of Operational Research, 236 (2014), 229-237.  doi: 10.1016/j.ejor.2013.11.020.

[23]

P. Ruíz-Canales and A. Rufián-Lizana, A characterization of weakly efficient points, Mathematical Programming, 68 (1995), 205-212.  doi: 10.1007/BF01585765.

[24]

R. E. Steuer and E. U. Choo, An interactive weighted {T}chebycheff procedure for multiple objective programming, Mathematical Programming, 26 (1983), 326-344.  doi: 10.1007/BF02591870.

[25]

Y. M. Xia, W. L. Zhang and K. Q. Zhao, Characterizations of improvement sets via quasi interior and applications in vector optimization, Optimization Letters, 10 (2016), 769-780.  doi: 10.1007/s11590-015-0897-0.

[26] P. L. Yu, Multiple-criteria Decision Making: Concepts, Techniques, and Extension, Plenum Press, New York, 1985.  doi: 10.1007/978-1-4684-8395-6.
[27]

K. Q. Zhao and X. M. Yang, $E$-Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752.  doi: 10.1080/02331934.2013.798321.

[28]

K. Q. Zhao, X. M. Yang and J. W. Peng, Weak {$E$}-optimal solution in vector optimization, Taiwanese Journal of Mathematics, 17 (2013), 1287–1302. doi: 10.11650/tjm.17.2013.2721.

[29]

C. Zopounidis and P. M. Pardalos, Handbook of Multicriteria Analysis, Springer, Berlin, 2010.

show all references

References:
[1]

F. Akbari, M. Ghaznavi and E. Khorram, A revised Pascoletti-Serafini scalarization method for multiobjective optimization problems, Journal of Optimization Theory and Applications, 178 (2018), 560-590.  doi: 10.1007/s10957-018-1289-2.

[2]

H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 71 (1979), 232-241.  doi: 10.1016/0022-247X(79)90226-9.

[3]

R. S. Burachik, C. Y. Kaya and M. M. Rizvi, A new scalarization technique to approximate Pareto fronts of problems with disconnected feasible sets, Journal of Optimization Theory and Applications, 162 (2014), 428-446.  doi: 10.1007/s10957-013-0346-0.

[4]

G. Y. Chen, Necessary conditions of nondominated solutions in multicriteria decision making, Journal of Mathematical Analysis and Applications, 104 (1984), 38-46.  doi: 10.1016/0022-247X(84)90027-1.

[5]

E. U. Choo and D. R. Atkins, Proper efficiency in nonconvex multicriteria programming, Mathematics of Operations Research, 8 (1983), 467-470. 

[6]

A. Chinchuluun and P. M. Pardalos, A survey of recent developments in multiobjective optimization, Annals of Operations Research, 154 (2007), 29-50.  doi: 10.1007/s10479-007-0186-0.

[7]

Y. Collette and P. Siarry, Multiobjective Optimization. Principles and Case Studies, , Springer, Berlin, 2003.

[8]

J. Dutta and C. Y. Kaya, A new scalarization and numerical method for constructing the weak pareto front of multi-objective optimization problems, Optimization, 60 (2011), 1091-1104.  doi: 10.1080/02331934.2011.587006.

[9]

M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.

[10]

M. Ehrgott and S. Ruzika, Improved $\varepsilon$-constraint method for multiobjective programming, Journal of Optimization Theory and Applications, 138 (2008), 375-396.  doi: 10.1007/s10957-008-9394-2.

[11]

A. Engau and M. M. Wiecek, Generating $\epsilon$-efficient solutions in multiobjective programming, European Journal of Operational Research, 177 (2007), 1566-1579.  doi: 10.1016/j.ejor.2005.10.023.

[12]

A. M. Geoffrion, Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications, 22 (1968), 618-630.  doi: 10.1016/0022-247X(68)90201-1.

[13]

A. Ghane-Kanafi and E. Khorram, A new scalarization method for finding the efficient frontier in non-convex multi-objective problems, Applied Mathematical Modelling, 39 (2015), 7483-7498.  doi: 10.1016/j.apm.2015.03.022.

[14]

B. A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization, Optimization, 62 (2013), 703-720.  doi: 10.1080/02331934.2012.668190.

[15]

S. M. Guu, N. J. Huang and J. Li, Scalarization approaches for set-valued vector optimization problems and vector variational inequalities, Journal of Mathematical Analysis and Applications, 356 (2009), 564-576.  doi: 10.1016/j.jmaa.2009.03.040.

[16]

X. X. Huang and X. Q. Yang, On characterizations of proper efficiency for nonconvex multiobjective optimization, Journal of Global Optimization, 23 (2002), 213-231.  doi: 10.1023/A:1016522528364.

[17]

R. Kasimbeyli, Z. K. Ozturk, N. Kasimbeyli and et al., Comparison of some scalarization methods in multiobjective optimization, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1875-1905.  doi: 10.1007/s40840-017-0579-4.

[18]

S. S. Kutateladze, Convex $\varepsilon$-programming, Dokl. Akad. Nauk SSSR, 245 (1979), 1048-1050. 

[19]

P. Loridan, $\epsilon$-solutions in vector minimization problems, Journal of Optimization Theory and Applications, 43 (1984), 265-276.  doi: 10.1007/BF00936165.

[20]

D. T. Luc, T. Q. Phong and M. Volle, Scalarizing functions for generating the weakly efficient solution set in convex multiobjective problems, SIAM Journal on Optimization, 15 (2005), 987-1001.  doi: 10.1137/040603097.

[21]

S. Mishra and M. Noor, Some nondifferentiable multiobjective programming problems, Journal of Mathematical Analysis and Applications, 316 (2006), 472-482.  doi: 10.1016/j.jmaa.2005.04.067.

[22]

N. Rastegar and E. Khorram, A combined scalarizing method for multiobjective programming problems, European Journal of Operational Research, 236 (2014), 229-237.  doi: 10.1016/j.ejor.2013.11.020.

[23]

P. Ruíz-Canales and A. Rufián-Lizana, A characterization of weakly efficient points, Mathematical Programming, 68 (1995), 205-212.  doi: 10.1007/BF01585765.

[24]

R. E. Steuer and E. U. Choo, An interactive weighted {T}chebycheff procedure for multiple objective programming, Mathematical Programming, 26 (1983), 326-344.  doi: 10.1007/BF02591870.

[25]

Y. M. Xia, W. L. Zhang and K. Q. Zhao, Characterizations of improvement sets via quasi interior and applications in vector optimization, Optimization Letters, 10 (2016), 769-780.  doi: 10.1007/s11590-015-0897-0.

[26] P. L. Yu, Multiple-criteria Decision Making: Concepts, Techniques, and Extension, Plenum Press, New York, 1985.  doi: 10.1007/978-1-4684-8395-6.
[27]

K. Q. Zhao and X. M. Yang, $E$-Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752.  doi: 10.1080/02331934.2013.798321.

[28]

K. Q. Zhao, X. M. Yang and J. W. Peng, Weak {$E$}-optimal solution in vector optimization, Taiwanese Journal of Mathematics, 17 (2013), 1287–1302. doi: 10.11650/tjm.17.2013.2721.

[29]

C. Zopounidis and P. M. Pardalos, Handbook of Multicriteria Analysis, Springer, Berlin, 2010.

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