v2010.10.26 - Convex Optimization

134 CHAPTER 2. CONVEX GEOMETRYwhich describe an intersection of four halfspaces in R m(m+1)/2 . Thatintersection creates the proper polyhedral cone K (2.12.1) whoseconstruction is illustrated in Figure 48. Drawn truncated is the boundaryof the positive semidefinite cone svec S 2 + and the bounding hyperplanessupporting K .Created by means of Geršgorin discs, K always belongs to the positivesemidefinite cone for any nonnegative value of p ∈ R m + . Hence any point inK corresponds to some positive semidefinite matrix A . Only the extremedirections of K intersect the positive semidefinite cone boundary in thisdimension; the four extreme directions of K are extreme directions of thepositive semidefinite cone. As p 1 /p 2 increases in value from 0, two extremedirections of K sweep the entire boundary of this positive semidefinite cone.Because the entire positive semidefinite cone can be swept by K , the systemof linear inequalities[Y T p1 ±psvec A 2 / √ ]2 00 ±p 1 / √ svec A ≽ 0 (252)2 p 2when made dynamic can replace a semidefinite constraint A≽0 ; id est, forgiven p where Y ∈ R m(m+1)/2×m2m−1butK = {z | Y T z ≽ 0} ⊂ svec S m + (253)svec A ∈ K ⇒ A ∈ S m + (254)∃p Y T svec A ≽ 0 ⇔ A ≽ 0 (255)In other words, diagonal dominance [202, p.349,7.2.3]A ii ≥m∑|A ij | , ∀i = 1... m (256)j=1j ≠ iis only a sufficient condition for membership to the PSD cone; but bydynamic weighting p in this dimension, it was made necessary andsufficient.In higher dimension (m > 2), boundary of the positive semidefinite coneis no longer constituted completely by its extreme directions (symmetric