v2007.11.26 - Convex Optimization

2.6. EXTREME, EXPOSED 77In other words, x ε is an extreme point of C if and only if x ε is not apoint relatively interior to any line segment in C . [270,2.10]Borwein & Lewis offer: [41,4.1.6] An extreme point of a convex set C isa point x ε in C whose relative complement C \x ε is convex.The set consisting of a single point C ={x ε } is itself an extreme point.2.6.0.0.2 Theorem. Extreme existence. [232,18.5.3] [20,II.3.5]A nonempty closed convex set containing no lines has at least one extremepoint.⋄2.6.0.0.3 Definition. Face, edge. [148,A.2.3]A face F of convex set C is a convex subset F ⊆ C such that everyclosed line segment x 1 x 2 in C , having a relatively interior point(x∈rel intx 1 x 2 ) in F , has both endpoints in F . The zero-dimensionalfaces of C constitute its extreme points. The empty set and C itselfare conventional faces of C . [232,18]All faces F are extreme sets by definition; id est, for F ⊆ C and allx 1 ,x 2 ∈ C \Fµ x 1 + (1 − µ)x 2 /∈ F , µ ∈ [0, 1] (136)A one-dimensional face of a convex set is called an edge.△Dimension of a face is the penultimate number of affinely independentpoints (2.4.2.3) belonging to it;dim F = sup dim{x 2 − x 1 , x 3 − x 1 , ... , x ρ − x 1 | x i ∈ F , i=1... ρ} (137)ρThe point of intersection in C with a strictly supporting hyperplaneidentifies an extreme point, but not vice versa. The nonempty intersection ofany supporting hyperplane with C identifies a face, in general, but not viceversa. To acquire a converse, the concept exposed face requires introduction: