v2009.01.01 - Convex Optimization

2.6. EXTREME, EXPOSED 83
2.6 Extreme, Exposed
2.6.0.0.1 Definition. Extreme point.
An extreme point x ε of a convex set C is a point, belonging to its closure C
[37,3.3], that is not expressible as a convex combination of points in C
distinct from x ε ; id est, for x ε ∈ C and all x 1 ,x 2 ∈ C \x ε
µx 1 + (1 − µ)x 2 ≠ x ε , µ ∈ [0, 1] (147)
In other words, x ε is an extreme point of C if and only if x ε is not a
point relatively interior to any line segment in C . [311,2.10]
Borwein & Lewis offer: [48,4.1.6] An extreme point of a convex set C is
a 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. [266,18.5.3] [25,II.3.5]
A nonempty closed convex set containing no lines has at least one extreme
point.
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2.6.0.0.3 Definition. Face, edge. [173,A.2.3]
A face F of convex set C is a convex subset F ⊆ C such that every
closed 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-dimensional
faces of C constitute its extreme points. The empty set and C itself
are conventional faces of C . [266,18]
All faces F are extreme sets by definition; id est, for F ⊆ C and all
x 1 ,x 2 ∈ C \F
µx 1 + (1 − µ)x 2 /∈ F , µ ∈ [0, 1] (148)
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A one-dimensional face of a convex set is called an edge.
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Dimension of a face is the penultimate number of affinely independent
points (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} (149)
ρ