A New Type of Directional Regularity for Mappings and Applications to Optimization

In this work we introduce and study a new type of directional regularity for mappings, making use of a minimal time function analyzed by the authors in a previous work. The corresponding directional triplet of linear openness, metric regularity, and Aubin property naturally appears, as shown by several examples. Then we devise a new directional Ekeland variational principle, which we use, along with other tools, to obtain necessary and sufficient conditions for directional regularity, formulated in terms of generalized differentiation objects. Finally, we apply the machinery developed before to the study of optimization problems with functional objectives.