v2004.06.19 - Convex Optimization

2.5. EXTREME, EXPOSED 472.5 Extreme, ExposedDefinition. Extreme point. An extreme point x ε of a convex set Cis a point, belonging to its closure C [33, §3.3], that is not expressible as aconvex combination of points in C distinct from x ε ; id est, for x ε ∈ C andall x 1 ,x 2 ∈ C \x ε ,µx 1 + (1 − µ)x 2 ≠ x ε , µ ∈ [0, 1] (100)△In other words, x ε is an extreme point of C if and only if x ε is not a pointrelatively interior to any line segment in C . [35, §2.10]The set consisting of a single point C ={x ε } is itself an extreme point.Theorem. Extreme existence. [30, §18.5.3] [23, §II.3.5] A nonemptyclosed convex set containing no lines has at least one extreme point. ⋄Definition. Face, edge. [29, §A.2.3]• A face F of convex set C is a convex subset F ⊆ C such that every closedline segment x 1 x 2 in C , having an interior-point x∈rel intx 1 x 2 in F ,has both endpoints in F . The zero-dimensional faces of C constituteits extreme points. The empty set and C itself are conventional facesof C . [30, §18]• All faces F are extreme sets by definition; id est, for F ⊆ C and allx 1 ,x 2 ∈ C \F ,µx 1 + (1 − µ)x 2 /∈ F , µ ∈ [0, 1] (101)• A one-dimensional face of a convex set is called an edge.△The dimension of a face is the penultimate number of affinely independentpoints (§2.3.2.3.1) 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} (102)ρ