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An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems. (English) Zbl 1128.49008
Summary: In this paper, the notion of gap functions is extended from scalar case to vector one. Then, gap functions and generalized functions for several kinds of vector equilibrium problems are shown. As an application, the dual problem of a class of optimization problems with a system of vector equilibrium constraints (in short, OP) is established, the concavity of the dual function, the weak duality of (OP) and the saddle point sufficient condition are derived by using generalized gap functions.

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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[1] Ansari Q.H., Chan W.K. and Yang X.Q. (2004). The system of vector quasi-equilibrium problems with applications. J. Global Optim. 29: 45–57 · Zbl 1073.90032 · doi:10.1023/B:JOGO.0000035018.46514.ca
[2] Ansari Q.H., Schaible S. and Yao J.C. (2002). The system of generalized vector equilibrium problems with applications.. J. Global Optim. 22: 3–16 · Zbl 1041.90069 · doi:10.1023/A:1013857924393
[3] Ansari Q.H. and Yao J.C. (1999). An existence result for generalized vector equilibrium problem. Appl. Math. Lett. 12: 53–56 · Zbl 1014.49008 · doi:10.1016/S0893-9659(99)00121-4
[4] Bianchi M., Hadjisavvas N. and Schaible S. (1997). Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92(3): 527–542 · Zbl 0878.49007 · doi:10.1023/A:1022603406244
[5] Blum E. and Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. The Math. Student 63: 123–145 · Zbl 0888.49007
[6] Chadli O., Wong N.C. and Yao J.C. (2003). Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117: 245–266 · Zbl 1141.49306 · doi:10.1023/A:1023627606067
[7] Chen G.Y., Goh C.J. and Yang X.Q. (2000). On gap functions for vector variational inequalities. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibrium, pp 55–72. Kluwer Academic Publishers, Dordrecht, Boston, London · Zbl 0997.49006
[8] Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems 541, Springer-Verlag, Berlin, Heidelberg (2005) · Zbl 1104.90044
[9] Chen G.Y. and Yang X.Q. (2002). Characterizations of variable domination structures via nonlinear scalarization. J. Optim. Theory Appl. 112(1): 97–110 · Zbl 0988.49005 · doi:10.1023/A:1013044529035
[10] Chen G.Y., Yang X.Q. and Yu H. (2005). A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Global Optim. 32(4): 451–466 · Zbl 1130.90413 · doi:10.1007/s10898-003-2683-2
[11] Ding X.P., Yao J.C. and Lin L.J. (2004). Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces. J. Math. Anal. Appl. 298: 398–410 · Zbl 1072.49005 · doi:10.1016/j.jmaa.2004.05.039
[12] Fang Y.P. and Huang N.J. (2004). Existence results for systems of strong implicit vector variational inequalities. Acta Math. Hungar. 103: 265–277 · Zbl 1060.49003 · doi:10.1023/B:AMHU.0000028828.52601.9e
[13] Fang Y.P. and Huang N.J. (2004). Vector equilibrium type problems with (S)+-conditions. Optimization 53: 269–279 · Zbl 1052.49009 · doi:10.1080/02331930410001712652
[14] Flores-Bazán Y.P. (2003). Existence theory for finite-dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77: 249–297 · Zbl 1053.90110 · doi:10.1023/A:1024971128483
[15] Gerth C. and Weidner P. (1990). Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67: 297–320 · Zbl 0692.90063 · doi:10.1007/BF00940478
[16] Giannessi F. Theorem of alternative, quadratic programs, and complementarity problems. In: Cottle R.W., Giannessi F., Lions J.L. (ed.) Variational Inequality and Complementarity Problems, pp. 151–186. John Wiley and Sons, Chichester, England
[17] Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibrium. Kluwer Academic Publishers, Dordrecht, Boston, London (2000) · Zbl 0952.00009
[18] Goh C.J. and Yang X.Q. (2002). Duality in Optimization and Variational Inequalities. Taylor and Francis, London · Zbl 1125.90059
[19] Göpfert A., Riahi H., Tammer C. and Zălinescu C. (2003). Variational Methods in Partially Ordered Spaces. Springer-Verlag, Berlin
[20] Hadjisavvas N. and Schaible S. (1998). From scalar to vector equilibrium problems in the quasimonotone case. J. Optim. Theory Appl. 96(2): 297–309 · Zbl 0903.90141 · doi:10.1023/A:1022666014055
[21] Huang N.J. and Gao C.J. (2003). Some generalized vector variational inequalities and complementarity problems for multivalued mappings. Appl. Math. Lett. 16: 1003–1010 · Zbl 1041.49009 · doi:10.1016/S0893-9659(03)90087-5
[22] Huang N.J., Li J. and Thompson H.B. (2003). Implicit vector equilibrium problems with applications. Math. Comput. Modelling 37: 1343–1356 · Zbl 1080.90086 · doi:10.1016/S0895-7177(03)90045-8
[23] Huang, N.J., Li, J., Yao, J.C.: Gap functions and existence of solutions for a system of vector equilibrium problems. J. Optim. Theory Appl. (in press) · Zbl 1146.49005
[24] Horst R., Pardalos P.M. and Thoai N.V. (1995). Introduce to Global Optimization. Kluwer Academic Publishers, Dordrecht, Boston, London · Zbl 0836.90134
[25] Isac G., Bulavski V.A. and Kalashnikov V.V. (2002). Complementarity, Equilibrium, Efficiency and Economics. Kluwer Academic Publishers, Dordrecht, Boston, London · Zbl 1081.90001
[26] Li J. and He Z.Q. (2005). Gap functions and existence of solutions to generalized vector variational inequalities. Appl. Math. Lett. 18(9): 989–1000 · Zbl 1079.49006 · doi:10.1016/j.aml.2004.06.029
[27] Li J. and Huang N.J. (2005). Implicit vector equilibrium problems via nonlinear scalarisation. Bulletin of the Australian Mathematical Society 72(1): 161–172 · Zbl 1081.49008 · doi:10.1017/S000497270003495X
[28] Li J., Huang N.J. and Kim J.K. (2003). On implicit vector equilibrium problems. J. Math. Anal. Appl. 283: 501–512 · Zbl 1137.90715 · doi:10.1016/S0022-247X(03)00277-4
[29] Li S.J., Teo K.L. and Yang X.Q. (2005). Generalized vector quasi-equilibrium problems. Math. Meth. Oper. Res. 61: 385–397 · Zbl 1114.90114 · doi:10.1007/s001860400412
[30] Lin L.J. and Chen H.L. (2005). The study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequalities problems. J. Global Optim. 32: 135–157 · Zbl 1079.90153 · doi:10.1007/s10898-004-2119-7
[31] Mastroeni G. (2003). Gap functions for equilibrium problems. J. Global Optim. 27: 411–426 · Zbl 1061.90112 · doi:10.1023/A:1026050425030
[32] Yang X.Q. (2003). On the gap functions of prevariational inequalities. J. Optim. Theory Appl. 116: 437–452 · Zbl 1027.49004 · doi:10.1023/A:1022422407705
[33] Yang X.Q. and Yao J.C. (2002). Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115: 407–417 · Zbl 1027.49003 · doi:10.1023/A:1020844423345
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