v2007.09.13 - Convex Optimization

174 CHAPTER 2. CONVEX GEOMETRYwhileK ∗ M = {y ∈ R n | X ∗† y ≽ 0, X ∗⊥T y = 0} (382)is the dual monotone cone halfspace-description.2.13.9.5 More pointed cone descriptions with equality conditionConsider pointed polyhedral cone K having a linearly independent set ofgenerators and whose subspace membership is explicit; id est, we are giventhe ordinary halfspace-descriptionK = {x | Ax ≽ 0, Cx = 0} ⊆ R n(246a)where A ∈ R m×n and C ∈ R p×n . This can be equivalently written in termsof nullspace of C and vector ξ :K = {Zξ ∈ R n | AZξ ≽ 0} (383)where R(Z ∈ R n×n−rank C ) ∆ = N(C) . Assuming (353) is satisfiedrankX ∆ = rank ( (AZ) † ∈ R n−rank C×m) = m − l = dim aff K ≤ n − rankC(384)where l is the number of conically dependent rows in AZ (2.10)that must be removed to make ÂZ before the cone tables becomeapplicable. 2.60 Then the results collected in the cone tables admit theassignment ˆX =(ÂZ)† ∆ ∈ R n−rank C×m−l , where Â∈ Rm−l×n , followed withlinear transformation by Z . So we get the vertex-description, for (ÂZ)†skinny-or-square full-rank,K = {Z(ÂZ)† b | b ≽ 0} (385)From this and (315) we get a halfspace-description of the dual coneK ∗ = {y ∈ R n | (Z T Â T ) † Z T y ≽ 0} (386)2.60 When the conically dependent rows are removed, the rows remaining must be linearlyindependent for the cone tables to apply.