v2010.10.26 - Convex Optimization

2.12. CONVEX POLYHEDRA 1512.12.2.2 Pointedness(2.7.2.1.2) [330,2.10] Assuming all generators constituting the columns ofX ∈ R n×N are nonzero, polyhedral cone K is pointed if and only if there isno nonzero a ≽ 0 that solves Xa=0; id est, ifffind asubject to Xa = 01 T a = 1a ≽ 0(292)is infeasible or iff N(X) ∩ R N + = 0 . 2.56 Otherwise, the cone will contain atleast one line and there can be no vertex; id est, the cone cannot otherwisebe pointed. Euclidean vector space R n or any halfspace are examples ofnonpointed polyhedral cone; hence, no vertex. This null-pointedness criterionXa=0 means that a pointed polyhedral cone is invariant to linear injectivetransformation.Examples of pointed polyhedral cone K include: the origin, any 0-basedray in a subspace, any two-dimensional V-shaped cone in a subspace, anyorthant in R n or R m×n ; e.g., nonnegative real line R + in vector space R .2.12.3 Unit simplexA peculiar subset of the nonnegative orthant with halfspace-descriptionS {s | s ≽ 0, 1 T s ≤ 1} ⊆ R n + (293)is a unique bounded convex full-dimensional polyhedron called unit simplex(Figure 53) having n + 1 facets, n + 1 vertices, and dimensiondim S = n (294)The origin supplies one vertex while heads of the standard basis [202][331] {e i , i=1... n} in R n constitute those remaining; 2.57 thus itsvertex-description:S= conv {0, {e i , i=1... n}}= { [0 e 1 e 2 · · · e n ]a | a T 1 = 1, a ≽ 0 } (295)2.56 If rankX = n , then the dual cone K ∗ (2.13.1) is pointed. (310)2.57 In R 0 the unit simplex is the point at the origin, in R the unit simplex is the linesegment [0,1], in R 2 it is a triangle and its relative interior, in R 3 it is the convex hull ofa tetrahedron (Figure 53), in R 4 it is the convex hull of a pentatope [373], and so on.