v2007.09.17 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 167from which a halfspace-description for the dual cone follows directly:By defining a matrixK ∗ = {y ∈ R n | X T y ≽ 0} (355)X ⊥ ∆ = basis N(X T ) (356)(a columnar basis for the orthogonal complement of R(X)), we can sayaff cone X = aff K = {x | X ⊥T x = 0} (357)meaning K lies in a subspace, perhaps R n . Thus we have ahalfspace-descriptionK = {x∈ R n | X † x ≽ 0, X ⊥T x = 0} (358)and from (272), a vertex-description 2.58K ∗ = { [X †T X ⊥ −X ⊥ ]b | b ≽ 0 } ⊆ R n (359)These results are summarized for a pointed polyhedral cone, havinglinearly independent generators, and its ordinary dual:2.13.9.2 Simplicial caseCone Table 1 K K ∗vertex-description X X †T , ±X ⊥halfspace-description X † , X ⊥T X TWhen a convex cone is simplicial (2.12.3), Cone Table 1 simplifies becausethen aff coneX = R n : For square X and assuming simplicial K such thatrank(X ∈ R n×N ) = N ∆ = dim aff K = n (360)we haveCone Table S K K ∗vertex-description X X †Thalfspace-description X † X T2.58 These descriptions are not unique. A vertex-description of the dual cone, for example,might use four conically independent generators for a plane (2.10.0.0.1) when only threewould suffice.