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A note on vector network equilibrium principles. (English) Zbl 1131.90010
Summary: An example is given to show that the necessary conditions of Theorem 4.5 [in G. Y. Chen et al. Math. Methods Oper. Res. 49, No. 2, 239–253 (1999; Zbl 0939.90014)] and Theorem 2.1 (i) [in X. Q. C. J. Goh and X. Q. Yang, Eur. J. Oper. Res. 116, 615–628 (1999; Zbl 1009.90093)] for (weak) vector equilibrium flows may not hold. New $\xi$-equilibrium and parametric equilibrium flows are introduced. As a result, necessary and sufficient conditions between a weak vector equilibrium flow and an $\xi$-equilibrium flow and between a vector equilibrium flow and a parametric equilibrium flow are established.
##### MSC:
 90B10 Network models, deterministic (optimization) 90B20 Traffic problems 91A40 Game-theoretic models
##### References:
 [1] Chen GY, Yen ND (1993) On the variational inequality model for network equilibrium. Internal Report 3.196 (724), Department of Mathematics, University of Pisa [2] Chen GY, Goh CJ, Yang XQ (1999) Vector network equilibrium problems and nonlinear scalarization methods. Math Methods Oper Res 49:239–253 [3] Gerth C, Weidner P (1990) Nonconvex separation theorems and some applications in vector optimization. J Optim Theor Appl 67:297–320 · Zbl 0692.90063 · doi:10.1007/BF00940478 [4] Goh CJ, Yang XQ (1999) Vector equilibrium problem and vector optimization. Eur J Oper Res 116:615–628 · Zbl 1009.90093 · doi:10.1016/S0377-2217(98)00047-2