On the Stability of Convex-valued Mappings and Their Relative Boundary and Extreme Points Set Mappings

This paper deals with the transmission of the main stability properties (lower and upper semicontinuity in Berge sense, and closedness) from a given closed--convex-valued mapping to its corresponding relative boundary and extreme point set mappings, and vice versa. The domain of the mappings considered in this paper are locally metrizable spaces and the images range on Euclidean spaces. Important examples of the class of mappings considered in this paper are the feasible set mapping and the optimal set mapping of convex optimization problems, for which the space of parameters is the result of perturbing a given nominal problem.