v2006.03.09 - Convex Optimization

2.11. WHEN EXTREME MEANS EXPOSED 1352.10.3 Utility of conic independencePerhaps the most useful application of conic independence is determinationof the intersection of closed convex cones from their halfspace-descriptions,or representation of the sum of closed convex cones from theirvertex-descriptions.⋂KiA halfspace-description for the intersection of any number of closedconvex cones K i can be acquired by pruning normals; specifically,only the conically independent normals from the aggregate of all thehalfspace-descriptions need be retained.∑KiGenerators for the sum of any number of closed convex cones K i canbe determined by retaining only the conically independent generatorsfrom the aggregate of all the vertex-descriptions.Such conically independent sets are not necessarily unique or minimal.2.11 When extreme means exposedFor any convex polyhedral set in R n having nonempty interior, distinctionbetween the terms extreme and exposed vanishes [210,2.4] [61,2.2] forfaces of all dimensions except n ; their meanings become equivalent as wesaw in Figure 11 (discussed in2.6.1.2). In other words, each and every faceof any polyhedral set (except the set itself) can be exposed by a hyperplane,and vice versa; e.g., Figure 12.Lewis [152,6] [133,2.3.4] claims nonempty extreme proper subsets andthe exposed subsets coincide for S n + ; id est, each and every face of the positivesemidefinite cone, whose dimension is less than the dimension of the cone,is exposed. A more general discussion of cones having this property can befound in [220]; e.g., the Lorentz cone (141) [15,II.A].