v2009.01.01 - Convex Optimization

92 CHAPTER 2. CONVEX GEOMETRY
Familiar examples of convex cones include an unbounded ice-cream cone
united with its interior (a.k.a: second-order cone, quadratic cone, circular
cone (2.9.2.5.1), Lorentz cone (confer Figure 40) [53, exmps.2.3 & 2.25]),
{[ ]
}
x
K l = ∈ R n × R | ‖x‖
t
l ≤ t , l=2 (160)
and any polyhedral cone (2.12.1.0.1); e.g., any orthant generated by
Cartesian half-axes (2.1.3). Esoteric examples of convex cones include
the point at the origin, any line through the origin, any ray having the
origin as base such as the nonnegative real line R + in subspace R , any
halfspace partially bounded by a hyperplane through the origin, the positive
semidefinite cone S M + (173), the cone of Euclidean distance matrices EDM N
(800) (6), any subspace, and Euclidean vector space R n .
2.7.2.1 cone invariance
More Euclidean bodies are cones, it seems, than are not. (confer Figures:
20, 28, 29, 30, 31, 32, 33, 35, 37, 44, 47, 50, 52, 53, 54, 55, 56, 57, 58,
114, 127, 150) This class of convex body, the convex cone, is invariant
to nonnegative scaling, linear and single- or many-valued inverse linear
transformation, their vector summation, Cartesian product, and intersection,
[266, p.22] but is not invariant to translation or projection: e.g., Figure 39.
2.7.2.1.1 Theorem. Cone intersection. [266,2,19]
The intersection of an arbitrary collection of convex cones is a convex cone.
Intersection of an arbitrary collection of closed convex cones is a closed
convex cone. [222,2.3] Intersection of a finite number of polyhedral cones
(2.12.1.0.1, Figure 44 p.128) is polyhedral.
⋄
The property pointedness is associated with a convex cone.
2.7.2.1.2 Definition. Pointed convex cone. (confer2.12.2.2)
A convex cone K is pointed iff it contains no line. Equivalently, K is not
pointed iff there exists any nonzero direction Γ ∈ K such that −Γ ∈ K .
[53,2.4.1] If the origin is an extreme point of K or, equivalently, if
K ∩ −K = {0} (161)
then K is pointed, and vice versa. [286,2.10]
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