v2009.01.01 - Convex Optimization
v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization
98 CHAPTER 2. CONVEX GEOMETRY ∂K ∗ K Figure 35: K is a pointed polyhedral cone having empty interior in R 3 (drawn truncated and in a plane parallel to the floor upon which you stand). K ∗ is a wedge whose truncated boundary is illustrated (drawn perpendicular to the floor). In this particular instance, K ⊂ int K ∗ (excepting the origin). Cartesian coordinate axes drawn for reference.
2.8. CONE BOUNDARY 99 2.8.1.1 extreme distinction, uniqueness An extreme direction is unique, but its vector representation Γ ε is not because any positive scaling of it produces another vector in the same (extreme) direction. Hence an extreme direction is unique to within a positive scaling. When we say extreme directions are distinct, we are referring to distinctness of rays containing them. Nonzero vectors of various length in the same extreme direction are therefore interpreted to be identical extreme directions. 2.34 The extreme directions of the polyhedral cone in Figure 20 (p.65), for example, correspond to its three edges. For any pointed polyhedral cone, there is a one-to-one correspondence of one-dimensional faces with extreme directions. The extreme directions of the positive semidefinite cone (2.9) comprise the infinite set of all symmetric rank-one matrices. [20,6] [170,III] It is sometimes prudent to instead consider the less infinite but complete normalized set, for M >0 (confer (206)) {zz T ∈ S M | ‖z‖= 1} (169) The positive semidefinite cone in one dimension M =1, S + the nonnegative real line, has one extreme direction belonging to its relative interior; an idiosyncrasy of dimension 1. Pointed closed convex cone K = {0} has no extreme direction because extreme directions are nonzero by definition. If closed convex cone K is not pointed, then it has no extreme directions and no vertex. [20,1] Conversely, pointed closed convex cone K is equivalent to the convex hull of its vertex and all its extreme directions. [266,18, p.167] That is the practical utility of extreme direction; to facilitate construction of polyhedral sets, apparent from the extremes theorem: 2.34 Like vectors, an extreme direction can be identified by the Cartesian point at the vector’s head with respect to the origin.
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98 CHAPTER 2. CONVEX GEOMETRY<br />
∂K ∗<br />
K<br />
Figure 35: K is a pointed polyhedral cone having empty interior in R 3<br />
(drawn truncated and in a plane parallel to the floor upon which you stand).<br />
K ∗ is a wedge whose truncated boundary is illustrated (drawn perpendicular<br />
to the floor). In this particular instance, K ⊂ int K ∗ (excepting the origin).<br />
Cartesian coordinate axes drawn for reference.
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