Directional stability theorem and directional metric regularity

We develop a new regularity concept, unifying metric regularity, Robinson's constraint qualification, and directional regularity. We present the directional stability theorem and the related concept of directional metric regularity. On one hand, our directional stability theorem immediately implies Robinson's stability theorem [Arutyunov, A. V 2005. Covering of nonlinear maps on cone in neighborhood of abnormal point. Math. Notes 77 447-460.] as a particular case, while on the other hand, our theorem easily implies various stability results under the directional regularity condition, widely used in sensitivity analysis. Some applications of this kind are also presented.