v2009.01.01 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 307
the first iteration of problem (693) is a 1-norm problem (456); id est,
minimize 〈t , 1〉
x∈R n , t∈R n
subject to Ax = b
−t ≼ x ≼ t
≡
minimize
x∈R n ‖x‖ 1
subject to Ax = b
(460)
Subsequent iterations of problem (693) engaging cardinality term 〈t , y〉 can
be interpreted as corrections to this 1-norm problem leading to a 0-norm
solution; vector y can be interpreted as a direction of search.
4.5.2.1 local convergence
As before (4.5.1.2), convex iteration (693) (467) always converges to a locally
optimal solution.
4.5.2.2 simple variations on a signed variable
Several useful equivalents to linear programs (693) (467) are easily devised,
but their geometrical interpretation is not as apparent: e.g., equivalent in
the limit ε→0 +
minimize 〈t , y〉
x∈R n , t∈R n
subject to Ax = b
(694)
−t ≼ x ≼ t
minimize
y∈R n 〈|x ⋆ |, y〉
subject to 0 ≼ y ≼ 1
y T 1 = n − k
(467)
We get another equivalent to linear programs (693) (467), in the limit, by
interpreting problem (460) as infimum to a vertex-description of the 1-norm
ball (Figure 61, Example 3.1.3.0.1, confer (459)):
minimize
a∈R 2n 〈a , y〉
subject to [A −A ]a = b
a ≽ 0
minimize
y∈R 2n 〈a ⋆ , y〉
subject to 0 ≼ y ≼ 1
y T 1 = 2n − k
(467)
(695)