Directional metric regularity of mappings and stability theorems

A stability theorem, based on the concept of directional matric regularity of mappings is described. Robinson's stability theorem can be used to derive results on the quantitative stability of the feasible set which play a central role in sensitivity analysis for optimization problems. The Robinson regularity condition, if violated, the underlying smooth mapping is not metrically regular. A mapping is said to be regular at a point in the direction where cone denotes the conical hull of a set. In the context of optimization problems, a condition is known as Gollan's regularity condition and it is extended to the general case and in parametric optimization, the condition is known as the directional regularity condition. An analysis performs along such feasible arcs yields the most accurate known quantitative sensitivity results in the case when the solution is Lipschitz stable.