## 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.

### MSC:

 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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### References:

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