v2007.09.13 - Convex Optimization

84 CHAPTER 2. CONVEX GEOMETRYXXFigure 26: Truncated nonconvex cone X = {x ∈ R 2 | x 1 ≥ x 2 , x 1 x 2 ≥ 0}.Boundary is also a cone. [181,2.4] Cartesian axes drawn for reference. Eachhalf (about the origin) is itself a convex cone.2.7.1 Cone definedA set X is called, simply, cone if and only ifΓ ∈ X ⇒ ζΓ ∈ X for all ζ ≥ 0 (143)where X denotes closure of cone X . An example of such a cone is theunion of two opposing quadrants; e.g., X = { x∈ R 2 | x 1 x 2 ≥0 } which is notconvex. [277,2.5] Similar examples are shown in Figure 22 and Figure 26.All cones can be defined by an aggregate of rays emanating exclusivelyfrom the origin (but not all cones are convex). Hence all closed cones containthe origin and are unbounded, excepting the simplest cone {0}. The emptyset ∅ is not a cone, but its conic hull is;cone ∅ ∆ = {0} (84)