In a finite-dimensional space this paper studied a class of optimization problems subject to variational inequality constraints. In terms of results in nonsmooth analysis first-order necessary optimality conditions are derived for the cases that the set-valued mapping of the solution of the variational inequality constraints is differentiable or not differentiable. Under certain conditions the optimization problem can be rewritten as a nondifferentiable optimization problem. It is thus claimed that a nondifferentiable optimization technique can be used for computing the solution. It is pointed out, however, that the first-order necessary condition obtained may not, in general, be sufficient.