-
Previous Article
Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation
- JIMO Home
- This Issue
-
Next Article
Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces
A combined scalarization method for multi-objective optimization problems
1. | Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China |
2. | School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China |
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.
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. 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. |
[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. Google Scholar |
[1] |
Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009 |
[2] |
Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130 |
[3] |
Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102 |
[4] |
Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020172 |
[5] |
Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109 |
[6] |
Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119 |
[7] |
Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021013 |
[8] |
Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020353 |
[9] |
Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020343 |
[10] |
Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 |
[11] |
Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 |
[12] |
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 |
[13] |
Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020049 |
[14] |
Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 |
[15] |
Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285 |
[16] |
Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 645-666. doi: 10.3934/dcdsb.2020262 |
[17] |
Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071 |
[18] |
Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329 |
[19] |
Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021025 |
[20] |
Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]