Projectionally Exposed Cones in R3

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Programming problems may be classified, on the basis of the objective function and types of constraints, as linear, nonlinear, discrete, integer, Boolean, etc. These programming problems represent special cases of the following more general abstract convex programming problem (ACPP): Find min{f{hook}(x):g(x)∈-K, x∈Ω}, where Ω⊆Rn is convex, K is a convex cone, and f,g are convex functions. Characterizations of optimality to the ACPP are of paramount importance in the investigation of optimization problems. A cone K in Rn is called projectionally exposed if for each face F of K there exists a projection PF of Rn such that PF(K) = F. In particular, it has been shown that when the constraint function g of the ACPP takes values in a projectionally exposed cone, then certain multipliers, associated with optimality, may be chosen from a smaller set. The projectionally exposed cones of R3 are completely characterized in this paper.