v2009.01.01 - Convex Optimization

120 CHAPTER 2. CONVEX GEOMETRY
Created by means of Geršgorin discs, K always belongs to the positive
semidefinite cone for any nonnegative value of p ∈ R m + . Hence any point in
K corresponds to some positive semidefinite matrix A . Only the extreme
directions of K intersect the positive semidefinite cone boundary in this
dimension; the four extreme directions of K are extreme directions of the
positive semidefinite cone. As p 1 /p 2 increases in value from 0, two extreme
directions of K sweep the entire boundary of this positive semidefinite cone.
Because the entire positive semidefinite cone can be swept by K , the system
of linear inequalities
[
Y T svec A =
∆ p1 ±p 2 / √ ]
2 0
0 ±p 1 / √ svec A ≽ 0 (224)
2 p 2
when made dynamic can replace a semidefinite constraint A≽0 ; id est, for
given p where Y ∈ R m(m+1)/2×m2m−1
but
K = {z | Y T z ≽ 0} ⊂ svec S m + (225)
svec A ∈ K ⇒ A ∈ S m + (226)
∃p Y T svec A ≽ 0 ⇔ A ≽ 0 (227)
In other words, diagonal dominance [176, p.349,7.2.3]
A ii ≥
m∑
|A ij | , ∀i = 1... m (228)
j=1
j ≠ i
is only a sufficient condition for membership to the PSD cone; but by
dynamic weighting p in this dimension, it was made necessary and
sufficient.
In higher dimension (m > 2), the boundary of the positive semidefinite
cone is no longer constituted completely by its extreme directions (symmetric
rank-one matrices); the geometry becomes complicated. How all the extreme
directions can be swept by an inscribed polyhedral cone, 2.40 similarly to the
foregoing example, remains an open question.
2.40 It is not necessary to sweep the entire boundary in higher dimension.