Hadamard well-posedness and stability in set optimization

In this paper, we introduce two kinds of Hadamard well-posedness for a set optimization problem by taking into consideration perturbations of objective function and a relationship between these two well-posedness is derived. Using the generalized Gerstewitz function, a sequence of scalar optimization problems have been defined and a convergence result is obtained. Sufficient conditions for Hadamard well-posedness for the set optimization problem are established by invoking these scalarization results. Finally, the stability of the convergence of minimal solution sets of the set optimization problem considered is discussed in terms of Painlevé-Kuratowski convergence.