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Vector network equilibrium problems and nonlinear scalarization methods

Author

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  • G. Y. Chen
  • C. J. Goh
  • X. Q. Yang

Abstract

The conventional equilibrium problem found in many economics and network models is based on a scalar cost, or a single objective. Recently, equilibrium problems based on a vector cost, or multicriteria, have received considerable attention. In this paper, we study a scalarization method for analyzing network equilibrium problems with vector-valued cost function. The method is based on a strictly monotone function originally proposed by Gerstewitz. Conditions that are both necessary and sufficient for weak vector equilibrium are derived, with the prominent feature that no convexity assumptions are needed, in contrast to other existing scalarization methods. Copyright Springer-Verlag Berlin Heidelberg 1999

Suggested Citation

  • G. Y. Chen & C. J. Goh & X. Q. Yang, 1999. "Vector network equilibrium problems and nonlinear scalarization methods," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(2), pages 239-253, April.
    • Handle: RePEc:spr:mathme:v:49:y:1999:i:2:p:239-253
    DOI: 10.1007/s001860050023
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    Cited by:

    1. Thai Doan Chuong, 2022. "Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization," Annals of Operations Research, Springer, vol. 311(2), pages 997-1015, April.
    2. Andrea Raith & Judith Wang & Matthias Ehrgott & Stuart Mitchell, 2014. "Solving multi-objective traffic assignment," Annals of Operations Research, Springer, vol. 222(1), pages 483-516, November.
    3. S. Li & M. Li, 2009. "Levitin–Polyak well-posedness of vector equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 125-140, March.
    4. Yunan Wu & Yuchen Peng & Long Peng & Ling Xu, 2012. "Super Efficiency of Multicriterion Network Equilibrium Model and Vector Variational Inequality," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 485-496, May.
    5. T. C. E. Cheng & Y. N. Wu, 2006. "A Multiproduct, Multicriterion Supply-Demand Network Equilibrium Model," Operations Research, INFORMS, vol. 54(3), pages 544-554, June.
    6. Gang Xiao & Hong Xiao & Sanyang Liu, 2011. "Scalarization and pointwise well-posedness in vector optimization problems," Journal of Global Optimization, Springer, vol. 49(4), pages 561-574, April.
    7. Jia-Wei Chen & Zhongping Wan & Yeol Cho, 2013. "Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 33-64, February.
    8. Dinh The Luc & Truong Thi Thanh Phuong, 2016. "Equilibrium in Multi-criteria Transportation Networks," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 116-147, April.
    9. Yu Han & Nan-jing Huang, 2018. "Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 679-695, June.
    10. S. Li & X. Yang & G. Chen, 2006. "A note on vector network equilibrium principles," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(2), pages 327-334, October.
    11. Li, S.J. & Teo, K.L. & Yang, X.Q., 2008. "A remark on a standard and linear vector network equilibrium problem with capacity constraints," European Journal of Operational Research, Elsevier, vol. 184(1), pages 13-23, January.
    12. Nguyen Van Hung & Vicente Novo & Vo Minh Tam, 2022. "Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone," Journal of Global Optimization, Springer, vol. 82(1), pages 139-159, January.
    13. J. H. Qiu & Y. Hao, 2010. "Scalarization of Henig Properly Efficient Points in Locally Convex Spaces," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 71-92, October.
    14. G. Y. Chen & X. Q. Yang, 2002. "Characterizations of Variable Domination Structures via Nonlinear Scalarization," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 97-110, January.
    15. Jiuping Xu & Guomin Fang & Zezhong Wu, 2016. "Network equilibrium of production, transportation and pricing for multi-product multi-market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(3), pages 567-595, December.

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