v2007.11.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 1672.13.9 Formulae, algorithm finding dual cone2.13.9.1 Pointed K , dual, X skinny-or-square full-rankWe wish to derive expressions for a convex cone and its ordinary dualunder the general assumptions: pointed polyhedral K denoted by its linearlyindependent extreme directions arranged columnar in matrix X such thatThe vertex-description is given:rank(X ∈ R n×N ) = N ∆ = dim aff K ≤ n (354)K = {Xa | a ≽ 0} ⊆ R n (355)from which a halfspace-description for the dual cone follows directly:By defining a matrixK ∗ = {y ∈ R n | X T y ≽ 0} (356)X ⊥ ∆ = basis N(X T ) (357)(a columnar basis for the orthogonal complement of R(X)), we can sayaff cone X = aff K = {x | X ⊥T x = 0} (358)meaning K lies in a subspace, perhaps R n . Thus we have ahalfspace-descriptionK = {x∈ R n | X † x ≽ 0, X ⊥T x = 0} (359)and from (273), a vertex-description 2.60K ∗ = { [X †T X ⊥ −X ⊥ ]b | b ≽ 0 } ⊆ R n (360)These results are summarized for a pointed polyhedral cone, havinglinearly independent generators, and its ordinary dual:Cone Table 1 K K ∗vertex-description X X †T , ±X ⊥halfspace-description X † , X ⊥T X T2.60 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.