v2010.10.26 - Convex Optimization

202 CHAPTER 2. CONVEX GEOMETRYwhere 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} (442)where R(Z ∈ R n×n−rank C ) N(C). Assuming (412) is satisfiedrankX rank ( (AZ) † ∈ R n−rank C×m) = m − l = dim aff K ≤ n − rankC(443)where l is the number of conically dependent rows in AZ which mustbe removed to make ÂZ before the Cone Tables become applicable.2.80Then results collected there admit assignment ˆX ( ÂZ) † ∈ R n−rank C×m−l ,where Â∈ Rm−l×n , followed with linear transformation by Z . So we get thevertex-description, for full-rank (ÂZ)† skinny-or-square,K = {Z(ÂZ)† b | b ≽ 0} (444)From this and (363) we get a halfspace-description of the dual coneK ∗ = {y ∈ R n | (Z T Â T ) † Z T y ≽ 0} (445)From this and Cone Table 1 (p.194) we get a vertex-description, (1873)K ∗ = {[Z †T (ÂZ)T C T −C T ]c | c ≽ 0} (446)Yet becauseK = {x | Ax ≽ 0} ∩ {x | Cx = 0} (447)then, by (314), we get an equivalent vertex-description for the dual coneK ∗ = {x | Ax ≽ 0} ∗ + {x | Cx = 0} ∗= {[A T C T −C T ]b | b ≽ 0}(448)from which the conically dependent columns may, of course, be removed.2.80 When the conically dependent rows (2.10) are removed, the rows remaining must belinearly independent for the Cone Tables (p.19) to apply.