多目标优化的广义Tchebycheff范数标量化

doi:10.15960/j.cnki.issn.1007-6093.2025.02.013

Abstract

Abstract:

Scalarization method is one of the basic research subjects for multi-objective optimization problems. In this paper, we first study the properties of generalized Tchebycheff norm, and obtain some strict monotonicity results on the non-negative quadrant. Furthermore, two kinds of scalarization results of weakly efficient solutions, efficient solutions, strictly efficient solutions and properly efficient solutions of multi-objective optimization problems are studied by using the properties of the generalized Tchebycheff norm. Moreover, we point out that under the assumption of convexity of the objective function, the scalarization studied in this paper is equivalent to the weighted scalarization.

Key words: multi-objective optimization, generalized Tchebycheff norm, strict monotonicity, scalarization, (weakly, strictly, properly) efficient solutions

CLC Number:

O221.6

Yuanmei XIA, Dandan XIA, Kequan ZHAO. Generalized Tchebycheff norm scalarization in multi-objective optimization[J]. Operations Research Transactions, 2025, 29(2): 175-183.

References 22

1	Coello-coello C A , Christiansen A D . Moses: A multiobjective optimization tool for engineering design[J]. Engineering Optimization, 1999, 31 (3): 337- 368. doi: 10.1080/03052159908941377
2	Samanlioglu F . A multi-objective mathematical model for the industrial hazardous waste location-routing problem[J]. European Journal of Operational Research, 2013, 226 (2): 332- 340. doi: 10.1016/j.ejor.2012.11.019
3	Zavala V M . A multiobjective optimization perspective on the stability of economic MPC[J]. IFAC-PapersOnLine, 2015, 48 (8): 974- 980. doi: 10.1016/j.ifacol.2015.09.096
4	Nguyen T H , Nong D , Paustian K . Surrogate-based multi-objective optimization of management options for agricultural landscapes using artificial neural networks[J]. Ecological Modelling, 2019, 400 (24): 1- 13.
5	Ehrgott M . Multicriteria Optimization[M]. Berlin: Springer, 2005.
6	Gao Y , Yang X M , Teo K L . Optimality conditions for approximate solutions of vector optimization problems[J]. Journal of Industrial and Management Optimization, 2011, 7 (2): 483- 496. doi: 10.3934/jimo.2011.7.483
7	Zhao K Q . Strong Kuhn-Tucker optimality in nonsmooth multiobjective optimization problems[J]. Pacific Journal of Optimization, 2015, 11 (3): 483- 494.
8	Yang X M . Generalized Preinvexity and Second Order Duality in Multiobjective Programming[M]. Singapore: Springer, 2018.
9	Gass S , Saaty T . The computational algorithm for the parametric objective function[J]. Naval Research Logistics Quarterly, 1955, 2 (1-2): 39- 45. doi: 10.1002/nav.3800020106
10	Geoffrion A M . Proper efficiency and the theory of vector maximization[J]. Journal of Mathematical Analysis and Applications, 1968, 22 (3): 618- 630. doi: 10.1016/0022-247X(68)90201-1
11	Haimes Y Y, Wismer D A, Lasdon L S. On a bicriterion formulation of the problems of integrated system identification and system optimization[J]. IEEE Transactions on Systems, Man, and Cybernetics, 1971, SMC-1(3): 296-297.
12	Ruíz-Canales P , Rufián-Lizana A . A characterization of weakly efficient points[J]. Mathematical Programming, 1995, 68 (1-3): 205- 212. doi: 10.1007/BF01585765
13	Ehrgott M , Ryan D M . Constructing robust crew schedules with bicriteria optimization[J]. Journal of Multi-Criteria Decision Analysis, 2002, 11 (3): 139- 150. doi: 10.1002/mcda.321
14	Ehrgott M , Ruzika S . Improved ε-constraint method for multiobjective programming[J]. Journal of Optimization Theory and Applications, 2008, 138 (3): 375- 396. doi: 10.1007/s10957-008-9394-2
15	Bowman V J. On the relationship of the Tchebycheff norm and the efficient frontier of multiple-criteria objectives[M]//Multiple Criteria Decision Making, Lecture Notes in Economics and Mathematical Systems, Berlin: Springer, 1976: 76-86.
16	Choo E U , Atkins D R . Proper efficiency in nonconvex multicriteria programming[J]. Mathematics of Operations Research, 1983, 8 (3): 467- 470. doi: 10.1287/moor.8.3.467
17	Steuer R E , Choo E U . An interactive weighted Tchebycheff procedure for multiple objective programming[J]. Mathematical Programming, 1983, 26 (3): 326- 344. doi: 10.1007/BF02591870
18	Pascoletti A , Serafini P . Scalarizing vector optimization problems[J]. Journal of Optimization Theory and Applications, 1984, 42 (4): 499- 524. doi: 10.1007/BF00934564
19	Akbari F , Ghaznavi M , Khorram E . A revised Pascoletti-Serafini scalarization method for multiobjective optimization problems[J]. Journal of Optimization Theory and Applications, 2018, 178 (3): 560- 590.
20	Xia Y M , Yang X M , Zhao K Q . A combined scalarization method for multi-objective optimization problems[J]. Journal of Industrial and Management Optimization, 2021, 17 (5): 2669- 2683. doi: 10.3934/jimo.2020088
21	赵克全, 杨新民, 夏远梅. 多目标优化问题的完全标量化[J]. 中国科学: 数学, 2021, 51 (2): 411- 424.
22	夏远梅. 多目标优化问题的标量化方法及其在机器学习中的应用[D]. 上海: 上海大学, 2020.

[1]	Xinyi LI, Ying GAO, Chunjie ZHAO. The smoothing method of the oriented distance function and its application [J]. Operations Research Transactions, 2024, 28(2): 117-130.
[2]	BAI Fusheng, CHEN Jiaoling. A clustering-based surrogate-assisted evolutionary algorithm for expensive multi-objective optimization [J]. Operations Research Transactions, 2022, 26(4): 31-42.
[3]	Xiuling WANG, Xunhua GONG. An existence theorem for efficient solutions of symmetric vector quasi-equilibrium problems [J]. Operations Research Transactions, 2021, 25(1): 73-80.
[4]	YU Ke, ZHANG Xiaoqing, ZHAO Kequan. Multi-objective optimization approach of the optimal performance allocation [J]. Operations Research Transactions, 2018, 22(4): 89-98.
[5]	LI Fei, TANG Liping, YANG Xinmin. A kind of cone-convexity for set-valued maps and its scalarization [J]. Operations Research Transactions, 2016, 20(4): 21-29.
[6]	TANG Liping, YANG Xinmin. A class of nonlinear scalarization theorem of vector optimization problems [J]. Operations Research Transactions, 2016, 20(3): 85-91.
[7]	ZHANG Wanli, XIA Yuanmei, ZHAO Kequan. Scalarization of weakly C(\varepsilon)-efficient solutions via quasi interior in vector optimization [J]. Operations Research Transactions, 2016, 20(2): 121-126.
[8]	LUO Guowang, PENG Yanfang, LIU Yanmin, HUANG Jianwen. Continuity of the solution set map to parametric weak vector equilibrium problems [J]. Operations Research Transactions, 2016, 20(1): 118-124.
[9]	GUO Hui, BAI Yanqin. Weak S-efficient solution of vector optimization [J]. Operations Research Transactions, 2016, 20(1): 54-60.
[10]	GUO Hui. A characterization of weakly (C,\varepsilon)-efficient solution of vector optimization via nonlinear scalarization [J]. Operations Research Transactions, 2015, 19(2): 105-110.
[11]	MU Lifeng, CAO Yan. A multi-objective method for supplier pre-selection of platform-based products [J]. Operations Research Transactions, 2015, 19(1): 31-44.
[12]	XIA Yuanmei, ZHAO Kequan. Nonlinear scalarization characterizations for \varepsilon-properly efficient solutions in vector optimization [J]. Operations Research Transactions, 2014, 18(4): 58-64.

Generalized Tchebycheff norm scalarization in multi-objective optimization

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

References 22

Related Articles 12

Recommended Articles

Metrics

Comments