Projectionally Exposed Cones

Abstract

Programming problems, based on the objective function and types of constraints, may be classified as linear, nonlinear, discrete, integer, and Boolean, just to name a few. These programming problems represent special cases of the more general Abstract Convex Programming Problem given by: Find $\operatorname{Min} \{ f(x):g(x) \in - K,x \in \Omega \} $ where $\Omega \subseteq \mathbb{R}^n $ is convex, K is a convex cone, and f, g are convex functions. Characterizations of optimality to the Abstract Convex Programming Problem are of paramount importance in the investigation of optimization problems. A cone K in $\mathbb{R}^n $ is called projectionally exposed if for each face F of K there exists a projection $P_F $ of $\mathbb{R}^n $ such that $P_F (K) = F$. In particular, it has been shown that when the constraint function g of the Abstract Convex Programming Problem takes values in a projectionally exposed cone, then certain multipliers, associated with optimality, may be chosen from a smaller set (see § 6 of [Borwein and Wolkowicz, J. Math. Anal. Appl., 83 (1981), pp. 495–530]). This suggests that a study of such cones is both applicable and intrinsically interesting. With this motivation, the authors have undertaken a project to characterize the cones of $\mathbb{R}^n $ which are projectionally exposed.