Strong Chip, Normality, and Linear Regularity of Convex Sets

We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a nite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at x is bounded away from 0 uniformly over all points in the intersection of these convex sets. 1. Introduction Brieey, this paper studies three forms of properties that can be imposed on a nite collection of closed convex sets in a Hilbert space: the strong CHIP, normality, and linear regularity. The concept of CHIP was rst introduced by Chui, Deutsch, and Ward 12] as a suucient condition for an uncon-strained reformulation of a constrained best approximation problem. Afterward, Deutsch, Li, and Ward 18] found that a stronger version of CHIP was actually needed for the reformulation of the constrained best approximation. In special cases, the reformulation leads to an unconstrained reformulation of a constrained optimization problem, which allows one to use various unconstrained optimization algorithms to solve constrained minimization problems 18]. Later, Deutsch 15] showed that the strong CHIP is a geometric version of the basic constraint qualiication for constrained optimization problems. The concept of linear regularity was rst introduced by Bauschke and Borwein as a key condition in establishing linear convergence rate of iterates generated by the cyclic projection algorithm for nding the projection from a point