Gap function for set-valued vector variational-like inequalities. (English) Zbl 1146.58009

Summary: Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem.


58E35 Variational inequalities (global problems) in infinite-dimensional spaces
90C22 Semidefinite programming
49J40 Variational inequalities
49J53 Set-valued and variational analysis
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[1] Yang, X.Q., Yao, J.C.: Gap functions and existence of solutions to set-valued vector variational inequalities. J. Optim. Theory Appl. 115, 407–417 (2002) · Zbl 1027.49003
[2] Lin, K.L., Yang, D.P., Yao, J.C.: Generalized vector variational inequalities. J. Optim. Theory Appl. 92, 117–125 (1997) · Zbl 0886.90157
[3] Daniilidis, A., Hadjisavvas, N.: Existence theorems for vector variational inequalities. Bull. Aust. Math. Soc. 54, 473–481 (1996) · Zbl 0887.49004
[4] Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problems. J. Math. Anal. Appl. 206, 42–58 (1997) · Zbl 0878.49006
[5] Chen, G.-Y.: Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992) · Zbl 0795.49010
[6] Chen, G.-Y., Cheng, G.M.: Vector variational inequalities and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–416. Springer, Berlin (1987)
[7] Parida, I., Sahoo, M., Kumar, A.: A variational-like inequality problem. Bull. Aust. Math. Soc. 39, 225–231 (1989) · Zbl 0649.49007
[8] Goh, C.J., Yang, X.Q.: Vector equilibrium problems and vector optimization. Eur. J. Oper. Res. 116, 615–628 (1999) · Zbl 1009.90093
[9] Ansari, Q.H., Yao, J.C.: On nondifferentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 475–488 (2000) · Zbl 0970.90092
[10] Chen, G.-Y., Goh, C.J., Yang, X.Q.: On a gap function for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 55–72. Kluwer Academic, Dordrecht (2000) · Zbl 0997.49006
[11] Ansari, Q.H., Yao, J.C.: Generalized variational-like inequalities and a gap function. Bull. Aust. Math. Soc. 59, 33–44 (1999) · Zbl 0928.49010
[12] Ding, X.P., Tarafdar, E.: Generalized vector variational-like inequalities with C x -{\(\eta\)}-pseudo-monotone set-valued mapping. In: Giannessi, F. (ed.) Vector variational inequalities and vector equilibria, pp. 125–140. Kluwer Academic, Dordrecht (2000) · Zbl 0991.49010
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