v2010.10.26 - Convex Optimization
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v2010.10.26 - Convex Optimization

v2010.10.26 - Convex Optimization v2010.10.26 - Convex Optimization

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108 CHAPTER 2. CONVEX GEOMETRY∂K ∗KFigure 41: K is a pointed polyhedral cone not full-dimensional in R 3 (drawntruncated in a plane parallel to the floor upon which you stand). Dual coneK ∗ is a wedge whose truncated boundary is illustrated (drawn perpendicularto the floor). In this particular instance, K ⊂ int K ∗ (excepting the origin).Cartesian coordinate axes drawn for reference.

2.8. CONE BOUNDARY 1092.8.1.1 extreme distinction, uniquenessAn extreme direction is unique, but its vector representation Γ ε is notbecause any positive scaling of it produces another vector in the same(extreme) direction. Hence an extreme direction is unique to within a positivescaling. When we say extreme directions are distinct, we are referring todistinctness of rays containing them. Nonzero vectors of various length inthe same extreme direction are therefore interpreted to be identical extremedirections. 2.36The extreme directions of the polyhedral cone in Figure 24 (p.71), forexample, correspond to its three edges. For any pointed polyhedral cone,there is a one-to-one correspondence of one-dimensional faces with extremedirections.The extreme directions of the positive semidefinite cone (2.9) comprisethe infinite set of all symmetric rank-one matrices. [21,6] [195,III] Itis sometimes prudent to instead consider the less infinite but completenormalized set, for M >0 (confer (235)){zz T ∈ S M | ‖z‖= 1} (187)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. [21,1]Conversely, pointed closed convex cone K is equivalent to the convex hullof its vertex and all its extreme directions. [307,18, p.167] That is thepractical utility of extreme direction; to facilitate construction of polyhedralsets, apparent from the extremes theorem:2.36 Like vectors, an extreme direction can be identified with the Cartesian point at thevector’s head with respect to the origin.

108 CHAPTER 2. CONVEX GEOMETRY∂K ∗KFigure 41: K is a pointed polyhedral cone not full-dimensional in R 3 (drawntruncated in a plane parallel to the floor upon which you stand). Dual coneK ∗ is a wedge whose truncated boundary is illustrated (drawn perpendicularto the floor). In this particular instance, K ⊂ int K ∗ (excepting the origin).Cartesian coordinate axes drawn for reference.

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