v2006.03.09 - Convex Optimization

2.8. CONE BOUNDARY 101Examples of proper cones are the positive semidefinite cone S M + in theambient space of symmetric matrices (2.9), the nonnegative real line R + invector space R , or any orthant in R n .2.8 Cone boundaryEvery hyperplane supporting a convex cone contains the origin. [123,A.4.2]Because any supporting hyperplane to a convex cone must therefore itself bea cone, then from the cone intersection theorem it follows:2.8.0.0.1 Lemma. Cone faces. [17,II.8]Each nonempty exposed face of a convex cone is a convex cone. ⋄2.8.0.0.2 Theorem. Proper-cone boundary.Suppose a nonzero point Γ lies on the boundary ∂K of proper cone K in R n .Then it follows that the ray {ζΓ | ζ ≥ 0} also belongs to ∂K . ⋄Proof. By virtue of its propriety, a proper cone guarantees the existenceof a strictly supporting hyperplane at the origin. [194, Cor.11.7.3] 2.23 Hencethe origin belongs to the boundary of K because it is the zero-dimensionalexposed face. The origin belongs to the ray through Γ , and the ray belongsto K by definition (137). By the cone faces lemma, each and every nonemptyexposed face must include the origin. Hence the closed line segment 0Γ mustlie in an exposed face of K because both endpoints do by Definition 2.6.1.3.1.That means there exists a supporting hyperplane ∂H to K containing 0Γ .So the ray through Γ belongs both to K and to ∂H . ∂H must thereforeexpose a face of K that contains the ray; id est,{ζΓ | ζ ≥ 0} ⊆ K ∩ ∂H ⊂ ∂K (147)2.23 Rockafellar’s corollary yields a supporting hyperplane at the origin to any convex conein R n not equal to R n .