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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  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 & Management Optimization, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[29]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[29]

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

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