A New Gap Function for Vector Variational Inequalities with an Application

We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.