v2010.10.26 - Convex Optimization

2.3. HULLS 65subset of the affine hull;conv C ⊆ aff C = aff C = aff C = aff conv C (87)Any closed bounded convex set C is equal to the convex hull of its boundary;C = conv ∂ C (88)conv ∅ ∅ (89)2.3.2.0.1 Example. Hull of rank-k projection matrices. [143] [287][11,4.1] [292,3] [239,2.4] [240] Convex hull of the set comprising outerproduct of orthonormal matrices has equivalent expression: for 1 ≤ k ≤ N(2.9.0.1)conv { UU T | U ∈ R N×k , U T U = I } = { A∈ S N | I ≽ A ≽ 0, 〈I , A〉=k } ⊂ S N +(90)This important convex body we call Fantope (after mathematician Ky Fan).In case k = 1, there is slight simplification: ((1619), Example 2.9.2.7.1)conv { UU T | U ∈ R N , U T U = 1 } = { A∈ S N | A ≽ 0, 〈I , A〉=1 } (91)In case k = N , the Fantope is identity matrix I . More generally, the set{UU T | U ∈ R N×k , U T U = I } (92)comprises the extreme points (2.6.0.0.1) of its convex hull. By (1463), eachand every extreme point UU T has only k nonzero eigenvalues λ and theyall equal 1 ; id est, λ(UU T ) 1:k = λ(U T U) = 1. So Frobenius’ norm of eachand every extreme point equals the same constant‖UU T ‖ 2 F = k (93)Each extreme point simultaneously lies on the boundary of the positivesemidefinite cone (when k < N ,2.9) and on the boundary of a hypersphereof dimension k(N −√k(1 k + 1) and radius − k ) centered at kI along2 2 N Nthe ray (base 0) through the identity matrix I in isomorphic vector spaceR N(N+1)/2 (2.2.2.1).Figure 22 illustrates extreme points (92) comprising the boundary of aFantope, the boundary of a disc corresponding to k = 1, N = 2 ; but thatcircumscription does not hold in higher dimension. (2.9.2.8)