Error Bound Characterizations of the Conical Constraint Qualification in Convex Programming
研究了Banach空间中凸不等式系统的锥约束条件的误差界刻画，给出了该条件成立的充要条件，并指出其仅适用于三种特殊情况。
This paper deals with error bound characterizations of the conical constraint qualification (CCQ) for convex inequality systems in a Banach space $X$. We establish necessary and sufficient conditions for a closed convex set $S$ defined by a convex function $g$ to have CCQ. These conditions are expressed in terms of the notion of error bound. Our results show that these characterizations hold in the following special cases: 1. $g$ is the maximum of a finite number of differentiable convex functions. 2. $S$ is closed convex and polyhedral. 3. The dimension of the subspace $\hbox{span}(S)$ is less than 2 and $g$ is positively homogeneous. We construct technical examples showing that these characterizations are limited to the three situations above. We introduce a new condition in terms of the gauge function which allows us to give an error bound characterization of convex nondifferentiable systems and to obtain as a direct consequence different characterizations of the concept of the strong conical hull intersection property for a finite collection of convex sets.