v2009.01.01 - Convex Optimization

64 CHAPTER 2. CONVEX GEOMETRY
2.3.2.0.5 Example. Convex hull of orthonormal matrices. [26,1.2]
Consider rank-k matrices U ∈ R n×k such that U T U = I . These are the
orthonormal matrices; a closed bounded submanifold, of all orthogonal
matrices, having dimension nk − 1 k(k + 1) [45]. Their convex hull is
2
expressed, for 1 ≤ k ≤ n
conv{U ∈ R n×k | U T U = I} = {X ∈ R n×k | ‖X‖ 2 ≤ 1}
= {X ∈ R n×k | ‖X T a‖ ≤ ‖a‖ ∀a∈ R n }
(92)
When k=n, matrices U are orthogonal. By Schur complement (A.4), the
spectral norm constraint in X can be expressed as a semidefinite constraint
[ ] I X
X T ≽ 0 (93)
I
because of equivalence X T X ≼ I ⇔ σ(X) ≤ 1 with singular values. (1453)
(1353) (1354)
2.3.3 Conic hull
In terms of a finite-length point list (or set) arranged columnar in X ∈ R n×N
(68), its conic hull is expressed
K ∆ = cone {x l , l=1... N} = coneX = {Xa | a ≽ 0} ⊆ R n (94)
id est, every nonnegative combination of points from the list. The conic hull
of any finite-length list forms a polyhedral cone [173,A.4.3] (2.12.1.0.1;
e.g., Figure 20); the smallest closed convex cone that contains the list.
By convention, the aberration [286,2.1]
Given some arbitrary set C , it is apparent
2.3.4 Vertex-description
cone ∅ ∆ = {0} (95)
conv C ⊆ cone C (96)
The conditions in (70), (78), and (94) respectively define an affine
combination, convex combination, and conic combination of elements from
the set or list. Whenever a Euclidean body can be described as some
hull or span of a set of points, then that representation is loosely called
a vertex-description.