ABSTRACT
As we have seen in Section 5.2, the variational inequality problem and complementarity problem are equivalent if the the underlying set is a convex cone. Karamardian [122, 123] and Saigal [187] considered the complementarity problem for set-valued maps, known as the generalized complementarity problem. If the underlying map in the formulation of the variational inequality problem (VIP) is a set-valued map, then the VIP is called a generalized variational inequality problem (GVIP). This problem was considered by Karamardian [122, 123] and Saigal [187] to prove the existence of a solution of a generalized complementarity problem. Basically, they extended Proposition 5.4 for set-valued maps and established an equivalence between a generalized complementarity problem and a generalized variational inequality problem under the condition that the underlying set is a convex cone. We have also seen in Section 5.2 that the solution set of a variational inequality problem coincides with the solution set of a minimization problem if the underlying function is convex and differentiable. However, if the function in the optimization problem is not necessarily differentiable, then the generalized variational inequality problem provides a necessary and sufficient condition for a solution of a minimization problem. A large number of papers on the existence of solutions of generalized variational inequality problems have appeared in the literature [8, 9, 42, 62, 65, 66, 75, 108, 109, 122, 123, 128, 135, 139, 146, 172, 187, 191, 192, 199, 205, 207, 210].
