We present definitions of types of variational convergence of finite-valued bifunctions defined on rectangular domains and establish characterizations of these convergences. In the introduction, we present the origins of the research on variational convergence and then we lead to the specific problem of this paper. The content of the paper consists of 3 parts: variational convergance of fucntion; variational convergance of bifunction; and characterizations of variational convergence of bifunction, this part is the main results of this paper. In section 2, we presented the definition of epi convergence and presented a basic property problem that will be used to extend and develop the next two sections. In section 3, we start to present a new definition, the definition of convergence epi / hypo, minsup and maxinf. To clearly understand of these new definitions we have provided comments (remarks) and some examples which reader can check these definitions. The above contents serve the main result of this paper will apply in part 4. Now, we will explain more detail for this part as follows. Firstly, variational convergence of bifunctions is characterized by the epi- and hypo-convergence of related unifunctions, which are slices sup- and inf-projections. The second characterization expresses the equivalence of variational convergence of bifunctions and the same convergence of the so-called proper bifunctions defined on the whole product spaces. In the third one, the geometric reformulation, we establish explicitly the interval of all the limits by computing formulae of the left- and right-end limit bifunctions, and this is necessary and sufficient conditions of the sequence bifunctions to attain epi / hypo, minsup and maxinf convergence.

In this paper, a general vector optimization problem with inequality constraints is considered, this topic is very popular and important model with a long research history in optimization. The generality of setting is mainly expressed in the following three factors. The underlying spaces being linear spaces without topology (except the decision space being additionally equipped with this structure in some results). The “orderings” in both objective and constraint spaces are defined by arbitrary nonempty sets (not necessarily convex cones). The problem data are nonsmooth mappings, i.e., they are not Fréchet differentiable. For this problem, the optimality conditions and Wolfe and Mond-Weir duality properties are investigated , which lie at the heart of optimization theory. These results are established for the three main and typical optimal solutions: (Pareto) minimal, weak minimal, and strong minimal solutions in both local and global considerations. The research define a type of Gateaux variation to play the role of a derivative. For optimality conditions, and introduce the concepts of on-set differentiable quasiconvexity for global solutions and sequential differentiable quasiconvexity for local ones. Furthermore, each of them is separated into type 1 for sufficient optimality conditions and type 2 for necessary ones. After obtaining optimality conditions, applying them to derive weak and strong duality relations for the above types of solutions following our duality schemes of the Wolfe and Mon-Weir types. Due to the complexity of the research subject: considerations of duality are different from that of optimality conditions, we have to design two more appropriate types of generalized quasiconvexity: scalar quasiconvexity for the weak solution and scalar strict convexity for the Pareto solution. So all the results are in terms of the aforementioned Gateaux variation and various types of generalized quasiconvexity. The results are remarkably different from the related known ones with some clear advantages in particular cases of applications.

We show first the definition of variational convergence of unifunctions and their basic variational properties. In the next section, we extend this variational convergence definition in case the functions which are defined on product two sets (bifunctions or bicomponent functions). We present the definition of variational convergence of bifunctions, icluding epi/hypo convergence, minsuplop convergnece and maxinf-lop convergence, defined on metric spaces. Its variational properties are also considered. In this paper, we concern on the properties of epi/hypo convergence to apply these results on optimization proplems in two last sections. Next we move on to the main results that are approximations of typical and important optimization related problems on metric space in terms of the types of variational convergence are equilibrium problems, and multiobjective optimization. When we applied to the finite dimensional case, some of our results improve known one.