Error bounds in mathematical programming
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
SIAM Journal on Optimization
SIAM Journal on Optimization
Constraint Qualifications for Semi-Infinite Systems of Convex Inequalities
SIAM Journal on Optimization
Metric Regularity and Constraint Qualifications for Convex Inequalities on Banach Spaces
SIAM Journal on Optimization
Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces
SIAM Journal on Optimization
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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 [11]. We construct an example to show that BCQ at a boundary point x does not ensure the metric regularity at x.