Characterizations of the nonemptiness and compactness for solution sets of convex set‐valued optimization problems

In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set‐valued optimization problem (with or without cone constraints) in which the decision space is finite‐dimensional. The characterizations are expressed in terms of the coercivity of some scalar set‐valued maps and the well‐posedness of the set‐valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set‐valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite‐dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set‐valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set‐valued optimization problems and the well‐posedness of the original problem.