Global well-posedness of set-valued optimization with application to uncertain problems

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).