v2010.10.26 - Convex Optimization

92 CHAPTER 2. CONVEX GEOMETRYthen vectorized matrix A is normal to a hyperplane (of dimension M 2 −1)that contains both vectorized nullspaces (each of whose dimension is M −ρ);vec A ⊥ vec basis N(A), vec basis N(A T ) ⊥ vec A (166)These vectorized subspace orthogonality relations represent a departure(absent T ) from fundamental subspace relations (137) stated at the outset.2.6 Extreme, Exposed2.6.0.0.1 Definition. Extreme point.An extreme point x ε of a convex set C is a point, belonging to its closure C[42,3.3], that is not expressible as a convex combination of points in Cdistinct from x ε ; id est, for x ε ∈ C and all x 1 ,x 2 ∈ C \x εµx 1 + (1 − µ)x 2 ≠ x ε , µ ∈ [0, 1] (167)△In 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 . [358,2.10]Borwein & Lewis offer: [55,4.1.6] 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. [307,18.5.3] [26,II.3.5]A nonempty closed convex set containing no lines has at least one extremepoint.⋄2.6.0.0.3 Definition. Face, edge. [199,A.2.3]Aface 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 . [307,18]All faces F are extreme sets by definition; id est, for F ⊆ C and allx 1 ,x 2 ∈ C \Fµx 1 + (1 − µ)x 2 /∈ F , µ ∈ [0, 1] (168)