v2007.09.13 - Convex Optimization

2.8. CONE BOUNDARY 93the same extreme direction are therefore interpreted to be identical extremedirections. 2.28The extreme directions of the polyhedral cone in Figure 15 (page 60), forexample, correspond to its three edges.The extreme directions of the positive semidefinite cone (2.9) comprisethe infinite set of all symmetric rank-one matrices. [18,6] [144,III] Itis sometimes prudent to instead consider the less infinite but completenormalized set, for M >0 (confer (193)){zz T ∈ S M | ‖z‖= 1} (156)The positive semidefinite cone in one dimension M =1, S + the nonnegativereal line, has one extreme direction belonging to its relative interior; anidiosyncrasy of dimension 1.Pointed closed convex cone K = {0} has no extreme direction becauseextreme directions are nonzero by definition.If closed convex cone K is not pointed, then it has no extreme directionsand no vertex. [18,1]Conversely, pointed closed convex cone K is equivalent to the convex hullof its vertex and all its extreme directions. [228,18, p.167] That is thepractical utility of extreme direction; to facilitate construction of polyhedralsets, apparent from the extremes theorem:2.8.1.1.1 Theorem. (Klee) Extremes. [245,3.6] [228,18, p.166](confer2.3.2,2.12.2.0.1) Any closed convex set containing no lines canbe expressed as the convex hull of its extreme points and extreme rays. ⋄It follows that any element of a convex set containing no lines maybe expressed as a linear combination of its extreme elements; e.g.,Example 2.9.2.4.1.2.8.1.2 GeneratorsIn the narrowest sense, generators for a convex set comprise any collectionof points and directions whose convex hull constructs the set.2.28 Like vectors, an extreme direction can be identified by the Cartesian point at thevector’s head with respect to the origin.