v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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130 CHAPTER 2. CONVEX GEOMETRY Barker & Carlson [20,1] call the extreme directions a minimal generating set for a pointed closed convex cone. A minimal set of generators is therefore a conically independent set of generators, and vice versa, 2.43 for a pointed closed convex cone. An arbitrary collection of n or fewer distinct extreme directions from pointed closed convex cone K ⊂ R n is not necessarily a linearly independent set; e.g., dual extreme directions (431) from Example 2.13.11.0.3. {≤ n extreme directions in R n } {l.i.} Linear dependence of few extreme directions is another convex idea that cannot be explained by a two-dimensional picture as Barvinok suggests [25, p.vii]; indeed, it only first comes to light in four dimensions! But there is a converse: [286,2.10.9] {extreme directions} ⇐ {l.i. generators of closed convex K} 2.10.2.0.1 Example. Vertex-description of halfspace H about origin. From n + 1 points in R n we can make a vertex-description of a convex cone that is a halfspace H , where {x l ∈ R n , l=1... n} constitutes a minimal set of generators for a hyperplane ∂H through the origin. An example is illustrated in Figure 45. By demanding the augmented set {x l ∈ R n , l=1... n + 1} be affinely independent (we want x n+1 not parallel to ∂H), then H = ⋃ (ζ x n+1 + ∂H) ζ ≥0 = {ζ x n+1 + cone{x l ∈ R n , l=1... n} | ζ ≥0} = cone{x l ∈ R n , l=1... n + 1} (257) a union of parallel hyperplanes. Cardinality is one step beyond dimension of the ambient space, but {x l ∀l} is a minimal set of generators for this convex cone H which has no extreme elements. 2.10.2.0.2 Exercise. Enumerating conically independent directions. Describe a nonpointed polyhedral cone in three dimensions having more than 8 conically independent generators. (confer Table 2.10.0.0.1) 2.43 This converse does not hold for nonpointed closed convex cones as Table 2.10.0.0.1 implies; e.g., ponder four conically independent generators for a plane (case n=2).

2.10. CONIC INDEPENDENCE (C.I.) 131 H x 2 x 3 0 ∂H Figure 45: Minimal set of generators X = [x 1 x 2 x 3 ]∈ R 2×3 (not extreme directions) for halfspace about origin; affinely and conically independent. x 1 2.10.3 Utility of conic independence Perhaps the most useful application of conic independence is determination of the intersection of closed convex cones from their halfspace-descriptions, or representation of the sum of closed convex cones from their vertex-descriptions. ⋂ Ki A halfspace-description for the intersection of any number of closed convex cones K i can be acquired by pruning normals; specifically, only the conically independent normals from the aggregate of all the halfspace-descriptions need be retained. ∑ Ki Generators for the sum of any number of closed convex cones K i can be determined by retaining only the conically independent generators from the aggregate of all the vertex-descriptions. Such conically independent sets are not necessarily unique or minimal.

2.10. CONIC INDEPENDENCE (C.I.) 131<br />

H<br />

x 2<br />

x 3<br />

0<br />

∂H<br />

Figure 45: Minimal set of generators X = [x 1 x 2 x 3 ]∈ R 2×3 (not extreme<br />

directions) for halfspace about origin; affinely and conically independent.<br />

x 1<br />

2.10.3 Utility of conic independence<br />

Perhaps the most useful application of conic independence is determination<br />

of the intersection of closed convex cones from their halfspace-descriptions,<br />

or representation of the sum of closed convex cones from their<br />

vertex-descriptions.<br />

⋂<br />

Ki<br />

A halfspace-description for the intersection of any number of closed<br />

convex cones K i can be acquired by pruning normals; specifically,<br />

only the conically independent normals from the aggregate of all the<br />

halfspace-descriptions need be retained.<br />

∑<br />

Ki<br />

Generators for the sum of any number of closed convex cones K i can<br />

be determined by retaining only the conically independent generators<br />

from the aggregate of all the vertex-descriptions.<br />

Such conically independent sets are not necessarily unique or minimal.

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