v2010.10.26 - Convex Optimization

194 CHAPTER 2. CONVEX GEOMETRY2.13.9 Formulae 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 (412)K = {Xa | a ≽ 0} ⊆ R n (413)from which a halfspace-description for the dual cone follows directly:By defining a matrixK ∗ = {y ∈ R n | X T y ≽ 0} (414)X ⊥ basis N(X T ) (415)(a columnar basis for the orthogonal complement of R(X)), we can sayaff cone X = aff K = {x | X ⊥T x = 0} (416)meaning K lies in a subspace, perhaps R n . Thus a halfspace-descriptionand a vertex-description 2.77 from (314)K = {x∈ R n | X † x ≽ 0, X ⊥T x = 0} (417)K ∗ = { [X †T X ⊥ −X ⊥ ]b | b ≽ 0 } ⊆ R n (418)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 T2.77 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, Figure 49) whenonly three would suffice.