v2009.01.01 - Convex Optimization

192 CHAPTER 2. CONVEX GEOMETRY
2.13.11.1 conic coordinate computation
The foregoing proof of the conic coordinates theorem is not constructive; it
establishes existence of dual extreme directions {Γj ∗ } that will reconstruct
a point x from its coordinates t ⋆ (x) via (435), but does not prescribe the
index set I . There are at least two computational methods for specifying
{Γj(i) ∗ } : one is combinatorial but sure to succeed, the other is a geometric
method that searches for a minimum of a nonconvex function. We describe
the latter:
Convex problem (P)
(P)
maximize t
t∈R
subject to x − tv ∈ K
minimize λ T x
λ∈R n
subject to λ T v = 1
λ ∈ K ∗ (D) (436)
is equivalent to definition (432) whereas convex problem (D) is its dual; 2.75
meaning, primal and dual optimal objectives are equal t ⋆ = λ ⋆T x assuming
Slater’s condition (p.256) is satisfied. Under this assumption of strong
duality, λ ⋆T (x − t ⋆ v)= t ⋆ (1 − λ ⋆T v)=0; which implies, the primal problem
is equivalent to
minimize λ ⋆T (x − tv)
t∈R
(P) (437)
subject to x − tv ∈ K
while the dual problem is equivalent to
minimize λ T (x − t ⋆ v)
λ∈R n
subject to λ T v = 1 (D) (438)
λ ∈ K ∗
Instead given coordinates t ⋆ (x) and a description of cone K , we propose
inversion by alternating solution of primal and dual problems
2.75 Form the Lagrangian associated with primal problem (P):
L(t, λ) = t + λ T (x − tv) = λ T x + t(1 − λ T v) ,
λ ≽
K ∗ 0
sup L(t, λ) = λ T x , 1 − λ T v = 0
t