In this study of domain optimization the functionals to be minimized are usually defined on various spaces. This causes an essential difficulty in establishing the existence for the optimal domains. In the existing research work an original domain optimization problem is normally transformed in order to get around this difficulty. However, some strong constraints on the admissible domains are usually needed to carry out such a transformation. This limits the problems that can be studied by this approach and causes extra complexity. In this paper existence of optimal domains is directly examined by introducing new concepts dealing with convergence of convex sets and functionals in various spaces. Several new concepts are first introduced to examine convergence of a sequence where elements are in different spaces. Then a new convergence concept is introduced for a sequence of functionals which are defined on various spaces. Compactness and lower- semicontinuity results with respect to this concept are established. These results are then applied to obtain some general existence results for optimal shape design for variational inequalities. It seems that this approach is applicable to a wide range of domain optimization problems.