v2010.10.26 - Convex Optimization

2.10. CONIC INDEPENDENCE (C.I.) 145Hx 2x 30∂HFigure 51: Minimal set of generators X = [x 1 x 2 x 3 ]∈ R 2×3 (not extremedirections) for halfspace about origin; affinely and conically independent.x 1An arbitrary collection of n or fewer distinct extreme directions frompointed closed convex cone K ⊂ R n is not necessarily a linearly independentset; e.g., dual extreme directions (483) 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 thatcannot be explained by a two-dimensional picture as Barvinok suggests[26, p.vii]; indeed, it only first comes to light in four dimensions! But thereis a converse: [330,2.10.9]{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 convexcone that is a halfspace H , where {x l ∈ R n , l=1... n} constitutes aminimal set of generators for a hyperplane ∂H through the origin. Anexample is illustrated in Figure 51. By demanding the augmented set