Characterizations of the strong basic constraint qualifications

In this paper, we characterize the general difference between strong basic constraint qualification (BCQ) and BCQ. For this purpose, we introduce a new measurement, the extent of a subdifferential, and show that for an inequality defined by a proper convex function f, the strong BCQ at a boundary point x of the solution set is equivalent to the extended BCQ plus the positivity of the extent of subdifferential at x. Applying the above characterization to the case when f is the maximum of finitely many differentiable convex functions, we show that the metric regularity at a boundary point x is equivalent to BCQ at every point in a “boundary-neighborhood” of x. In addition, we provide an answer to the open question proposed by Zheng and Ng. We construct an example to show that BCQ at a boundary point x does not ensure the metric regularity at x.