v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 181
2.13.9.5 More pointed cone descriptions with equality condition
Consider pointed polyhedral cone K having a linearly independent set of
generators and whose subspace membership is explicit; id est, we are given
the ordinary halfspace-description
K = {x | Ax ≽ 0, Cx = 0} ⊆ R n
(259a)
where A ∈ R m×n and C ∈ R p×n . This can be equivalently written in terms
of nullspace of C and vector ξ :
K = {Zξ ∈ R n | AZξ ≽ 0} (396)
where R(Z ∈ R n×n−rank C ) ∆ = N(C) . Assuming (366) is satisfied
rankX ∆ = rank ( (AZ) † ∈ R n−rank C×m) = m − l = dim aff K ≤ n − rankC
(397)
where l is the number of conically dependent rows in AZ (2.10)
that must be removed to make ÂZ before the cone tables become
applicable. 2.71 Then the results collected in the cone tables admit the
assignment ˆX =(ÂZ)† ∆ ∈ R n−rank C×m−l , where Â∈ Rm−l×n , followed with
linear transformation by Z . So we get the vertex-description, for (ÂZ)†
skinny-or-square full-rank,
K = {Z(ÂZ)† b | b ≽ 0} (398)
From this and (327) we get a halfspace-description of the dual cone
K ∗ = {y ∈ R n | (Z T Â T ) † Z T y ≽ 0} (399)
From this and Cone Table 1 (p.174) we get a vertex-description, (1749)
Yet because
K ∗ = {[Z †T (ÂZ)T C T −C T ]c | c ≽ 0} (400)
K = {x | Ax ≽ 0} ∩ {x | Cx = 0} (401)
then, by (284), we get an equivalent vertex-description for the dual cone
K ∗ = {x | Ax ≽ 0} ∗ + {x | Cx = 0} ∗
= {[A T C T −C T ]b | b ≽ 0}
from which the conically dependent columns may, of course, be removed.
(402)
2.71 When the conically dependent rows are removed, the rows remaining must be linearly
independent for the cone tables to apply.