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Vector variational inequality as a tool for studying vector optimization problems. (English) Zbl 0956.49007
The aim of the paper is to show that the vector variational inequality (VVI) can be an efficient tool for studying vector optimization problems. It is shown that the necessary condition for a weak efficient point of a vector optimization problem with differentiable functions is a VVI. This observation suggests a way to use existing results on variational inequalities for vector optimization problems. Various basic facts on the class of strongly monotone VVI are established: Connectedness and compactness of the solution set. Furthermore, a Hölder-Lipschitz continuity property of the solution set of parametric problems is studied. From this result the authors derive interesting information about vector optimization problems with $$\varrho$$-convex functions. Finally, they discuss a useful example of a strongly monotone VVI whose solution set is not a singleton.

##### MSC:
 49J40 Variational inequalities 90C29 Multi-objective and goal programming 90C25 Convex programming 90C31 Sensitivity, stability, parametric optimization
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##### References:
 [1] F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, In: R.W. Cottle, F. Giannessi, J.-L. Lions (Eds.), Variational Inequality and Complementarity Problems, pp. 151-186. · Zbl 0484.90081 [2] Chen, G.Y., Existence of solutions for a vector variational inequalityan extension of the hartman – stampacchia theorem, J. optim. theory appl., 74, 445-456, (1992) · Zbl 0795.49010 [3] G.Y. Chen, G.M. Cheng, Vector variational inequality and vector optimization, Lecture Notes in Economics and Mathematical Systems, vol. 285, Springer, Berlin (1988) pp. 408-416. [4] Chen, G.Y.; Craven, B.D., A vector variational inequality and optimization over an efficient set, ZOR-meth. models oper. res., 34, 1-12, (1990) · Zbl 0693.90091 [5] Chen, G.Y.; Yang, X.Q., The complementarity problems and their equivalence with the weak minimal element in ordered spaces, J. math. anal. appl., 153, 136-158, (1990) [6] Yang, X.Q., Vector complementarity and minimal element problems, J. optim. theory appl., 77, 483-495, (1993) · Zbl 0796.49014 [7] Yang, X.Q., Vector variational inequality and its duality, Nonlinear anal., 21, 869-877, (1993) · Zbl 0809.49009 [8] Siddiqi, A.H.; Ansari, Q.H.; Khaliq, A., On vector variational inequalities, J. optim. theory appl., 84, 171-180, (1995) · Zbl 0827.47050 [9] Lee, G.M.; Kim, D.S.; Lee, B.S.; Cho, S.J., Generalized vector variational inequality and fuzzy extension, Appl. math. lett., 6, 47-51, (1993) · Zbl 0804.49004 [10] Lee, G.M.; Kim, D.S.; Lee, B.S., Some existence theorems for generalized vector variational inequalities, Bull. Korean math. soc., 32, 343-348, (1995) · Zbl 0844.49007 [11] Lee, G.M.; Kim, D.S.; Lee, B.S., On vector quasivariational-like inequality, Bull. Korean math. soc., 33, 45-55, (1996) · Zbl 0864.47041 [12] Lee, G.M.; Kim, D.S.; Lee, B.S., Generalized vector variational inequality, Appl. math. lett., 9, 39-42, (1996) · Zbl 0862.49014 [13] Sawaragi, Y.; Nakayama, H.; Tanino, T., Theory of multiobjective optimization, (1985), Academic Press New York · Zbl 0566.90053 [14] D.T. Luc Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319, Springer-Verlag, Berlin, 1989. [15] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (1980), Academic Press New York · Zbl 0457.35001 [16] J.-P. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983) 231-259. · Zbl 0526.90077 [17] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton · Zbl 0229.90020 [18] Sach, P.H.; Yen, N.D.; Craven, B.D., Generalized invexity for multifunctions and duality theories, Numer. funct. anal. optim., 15, 131-153, (1994) · Zbl 0808.90129 [19] Yen, N.D., Hölder continuity of solutions to a parametric variational inequality, Appl. math. optim., 31, 245-255, (1995) · Zbl 0821.49011 [20] J.-P. Aubin Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984) 87-111. [21] Dafermos, S., Sensitivity analysis in variational inequalities, Math. oper. res., 13, 421-434, (1988) · Zbl 0674.49007 [22] Yen, N.D., Stability of the solution set of perturbed nonsmooth inequality systems and application, J. optim. theory appl., 93, 1, 199-225, (1997) · Zbl 0901.90168 [23] Tanino, T., Sensitivity analysis in multiobjective optimization, J. optim. theory appl., 56, 479-499, (1988) · Zbl 0619.90073 [24] Tanino, T., Stability and sensitivity analysis in convex vector optimization, SIAM J. control optim., 26, 521-536, (1988) · Zbl 0654.49011 [25] Tanino, T., Stability and sensitivity analysis in multiobjective nonlinear programming, Annals of oper. res., 27, 97-114, (1990) · Zbl 0719.90065
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