Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces. (English) Zbl 1143.58007

Authors’ abstract: We introduce and discuss the notion of \(\varepsilon\)-solutions of vector variational inequalities. Using convex analysis and nonsmooth analysis, we provide some sufficient conditions and necessary conditions for a point to be an \(\varepsilon\)-solution of vector variational inequalities.


58E35 Variational inequalities (global problems) in infinite-dimensional spaces
58E17 Multiobjective variational problems, Pareto optimality, applications to economics, etc.
90C46 Optimality conditions and duality in mathematical programming
Full Text: DOI


[1] Chen, G.-Y., Craven, B. D.: Approximate dual and approximate vector variational inequality for multiobjective optimization. J. Austral. Math. Soc. Ser. A 47, 418–423 (1989) · Zbl 0693.90089 · doi:10.1017/S1446788700033139
[2] Clarke, F.H.: Optimization and Nonsmooth Analysis. Les publications CRM, Montreal, Canada, (1989) · Zbl 0727.90045
[3] Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Mathematical theories. Nonconvex Optimization and its Applications, 38. Kluwer Academic Publishers, Dordrecht, (2000) · Zbl 0952.00009
[4] Goh, C.J., Yang, X.Q.: On scalarization methods for vector variational inequalities. In: by F. Giannessi,(ed) Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dordrecht/Boston/London, 217–232 (2000) · Zbl 0991.49014
[5] Jeyakumar, V.: Convexlike alternative theorems and mathematical programming. Optimization 16, 643–652 (1985) · Zbl 0581.90079 · doi:10.1080/02331938508843061
[6] Rong, W.D.: Epsilon-approximate solutions to vector optimization problems and vector variational inequalities. (Chinese) Nei Monggol Daxue Xuebao Ziran Kexue 23, 5130–518 (1992) · Zbl 1332.90258
[7] Ward, D.E., Lee, G.M.: On relations between vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 113, 583–596 (2002) · Zbl 1022.90024 · doi:10.1023/A:1015364905959
[8] Yang, X.Q.: Generalized convex functions and vector variational inequalities. J. Optim. Theory Appl. 79, 563–580 (1993) · Zbl 0797.90085 · doi:10.1007/BF00940559
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