V Vetrivel

Well-posedness for optimization problems is a well-known notion and has been studied extensively for scalar, vector, and set-valued optimization problems. For the set-valued case, there are many subdivisions: firstly in terms of pointwise notion and global notion and secondly in terms of the solution concepts, like the vector approach, the set-relation approach, etc. Various definitions of pointwise well-posedness for a set-valued optimization problem in the set-relation approach have been proposed in the literature. Here we do a comparative study and suggest modifications in some existing results. We also introduce a new pointwise well-posedness and discuss its properties and connection with others.

Well-posedness of optimization problems is a well explored topic in scalar, vector, and set-valued optimization literature. Pointwise well-posedness and global well-posedness are two of the major classes of well-posedness in multi-objective and set-valued optimization problems. For the set-valued case, there is further subdivision in terms of the various solution approaches, namely the vector approach, the set-relation approach etc. Our focus here is only in the set-relation approach. Many notions of pointwise well-posedness for a set-valued optimization problem are available. However, those have been defined only for minimal solutions, whereas the notions of global well-posedness have been studied for both minimal and weak minimal solutions. In this paper, therefore, we introduce some notions of pointwise well-posedness for a set-valued optimization problem corresponding to a weak minimal solution and discuss various properties and examples.

Well-posedness for optimization problems is a well-known notion and has been studied extensively for scalar, vector and set-valued optimization problems. There is a broad classification in terms of pointwise and global well-posedness notions in vector and set-valued optimization problems. We have focused on global well-posedness for set-valued optimization problems in this paper. A number of notions of global well-posedness for set-valued optimization problems already exist in the literature. However, we found equivalence between some existing notions of global well-posedness for set-valued optimization problems and also found scope of improving and extending the research in that field. That has been the first aim of this paper. On the other hand, robust approach towards uncertain optimization problems is another growing area of research. The well-posedness for the robust counterparts have been explored in very few papers, and that too only in the scalar and vector cases (see (Anh et al. in Ann Oper Res 295(2):517–533, 2020), (Crespi et al. in Ann Oper Res 251(1–2):89–104, 2017)). Therefore, the second aim of this paper is to study some global well-posedness properties of the robust formulation of uncertain set-valued optimization problems that generalize the concept of the well-posedness of robust formulation of uncertain vector optimization problems as discussed in Anh et al. (Ann Oper Res 295(2):517–533, 2020), Crespi et al. (Ann Oper Res 251(1–2):89–104, 2017).

We extend the concept of well-posedness to the split equilibrium problem and establish Furi–Vignoli-type characterizations for the well-posedness. We prove that the well-posedness of the split equilibrium problem is equivalent to the existence and uniqueness of its solution under certain assumptions on the bifunctions involved. We also characterize the generalized well-posedness of the split equilibrium problem via the Kuratowski measure of noncompactness. We illustrate our theoretical results by several examples.