v2010.10.26 - Convex Optimization

106 CHAPTER 2. CONVEX GEOMETRY2.7.2.2.1 Definition. Proper cone: a cone that ispointedclosedconvexfull-dimensional.△A proper cone remains proper under injective linear transformation.[90,5.1] Examples of proper cones are the positive semidefinite cone S M + inthe ambient space of symmetric matrices (2.9), the nonnegative real line R +in vector space R , or any orthant in R n , and the set of all coefficients ofunivariate degree-n polynomials nonnegative on interval [0, 1] [61, exmp.2.16]or univariate degree-2n polynomials nonnegative over R [61, exer.2.37].2.8 Cone boundaryEvery hyperplane supporting a convex cone contains the origin. [199,A.4.2]Because any supporting hyperplane to a convex cone must therefore itself bea cone, then from the cone intersection theorem (2.7.2.1.1) it follows:2.8.0.0.1 Lemma. Cone faces. [26,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 existenceof a strictly supporting hyperplane at the origin. [307, cor.11.7.3] 2.32 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 (174). 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 2.6.1.4.1.2.32 Rockafellar’s corollary yields a supporting hyperplane at the origin to any convex conein R n not equal to R n .