Abadie's constraint qualification, metric regularity, and error bounds for differentiable convex inequalities

In this paper we study differentiable convex inequalities and prove that metric regularity and Abadie's constraint qualification (CQ) are equivalent for such inequalities. For convex quadratic inequalities, we show that metric regularity, the existence of a global error bound, and Abadiel's CQ are mutually equivalent. As a consequence, we derive two new characterizations of weak sharp minima of a convex quadratic programming problem.