v2004.06.19 - Convex Optimization

2.2. HULLS 35Figure 2.3: A simplicial cone (§2.7.3.0.1) in R 3 whose boundary is drawntruncated; constructed using A∈ R 3×3 and C = 0 in (159). By the mostfundamental definition of a cone (§2.6.1), the entire boundary can be constructedfrom an aggregate of rays emanating exclusively from the origin.The extreme directions are the directions of the three edges (§2.5); they areconically and linearly independent for this cone. Because this set is polyhedral,the exposed directions are in one-to-one correspondence with theextreme directions; there are only three.2.2.2.1.1 Example. Convex hull of outer product. [51, §3] [52, §2.4]conv { XX T | X ∈ R n×k , X T X = I } = {A∈ S n | I ≽ A ≽ 0, 〈I , A 〉= k}(62)2.2.3 Conic hullIn terms of a finite-length point list (or set) arranged columnar in X ∈ R n×N(54), its conic hull is expressedK ∆ = cone {x l , l=1... N} = coneX = {Xa | a ≽ 0} ⊆ R n (63)The conic hull of any list forms a polyhedral cone [29, §A.4.3] (§2.7.1; e.g.,Figure 2.3); the smallest closed convex cone that contains the list.