v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 193
N∑
minimize Γ ∗T
x∈R n i (x − t ⋆ iΓ i )
i=1
subject to x − t ⋆ iΓ i ∈ K ,
i=1... N
(439)
N∑
minimize Γ ∗T
Γ ∗ i (x ⋆ − t ⋆ iΓ i )
i ∈Rn i=1
subject to Γ ∗T
i Γ i = 1 ,
Γ ∗ i ∈ K ∗ ,
i=1... N
i=1... N
(440)
where dual extreme directions Γ ∗ i are initialized arbitrarily and ultimately
ascertained by the alternation. Convex problems (439) and (440) are
iterated until convergence which is guaranteed by virtue of a monotonically
nonincreasing real sequence of objective values. Convergence can be fast.
The mapping t ⋆ (x) is uniquely inverted when the necessarily nonnegative
objective vanishes; id est, when Γ ∗T
i (x ⋆ − t ⋆ iΓ i )=0 ∀i. Here, a zero
objective can occur only at the true solution x . But this global optimality
condition cannot be guaranteed by the alternation because the common
objective function, when regarded in both primal x and dual Γ ∗ i variables
simultaneously, is generally neither quasiconvex or monotonic.
Conversely, a nonzero objective at convergence is a certificate that
inversion was not performed properly. A nonzero objective indicates that
the global minimum of a multimodal objective function could not be found
by this alternation. That is a flaw in this particular iterative algorithm for
inversion; not in theory. 2.76 A numerical remedy is to reinitialize the Γ ∗ i to
different values.
2.76 The Proof 2.13.11.0.5 that suitable dual extreme directions {Γj ∗ } always exist means
that a global optimization algorithm would always find the zero objective of alternation
(439) (440); hence, the unique inversion x. But such an algorithm can be combinatorial.