v2009.01.01 - Convex Optimization

262 CHAPTER 4. SEMIDEFINITE PROGRAMMING
an identical solution were the norm in problem (614) instead the 1-norm or
2-norm; id est, the two problems
(614)
minimize ‖x‖ 0
x
subject to Ax = b
x i ∈ {0, 1} ,
i=1... n
=
minimize ‖x‖ 1
x
subject to Ax = b
x i ∈ {0, 1} ,
(615)
i=1... n
are the same. The Boolean constraint makes the 1-norm problem nonconvex.
Given data 4.12
A =
⎡
−1 1 8 1 1 0
⎢
1 1 1
⎣ −3 2 8
2 3
−9 4 8
1
4
1
9
− 1 2 3
1
− 1 4 9
⎤
⎥
⎦ , b =
the obvious and desired solution to the problem posed,
⎡
⎢
⎣
1
1
2
1
4
⎤
⎥
⎦ (616)
x ⋆ = e 4 ∈ R 6 (617)
has norm ‖x ⋆ ‖ 2 =1 and minimum cardinality; the minimum number of
nonzero entries in vector x . The Matlab backslash command x=A\b ,
for example, finds
⎡
x M
=
⎢
⎣
2
128
0
5
128
0
90
128
0
⎤
⎥
⎦
(618)
having norm ‖x M
‖ 2 = 0.7044 .
id est, an optimal solution to
Coincidentally, x M
is a 1-norm solution;
minimize ‖x‖ 1
x
subject to Ax = b
(460)
4.12 This particular matrix A is full-rank having three-dimensional nullspace (but the
columns are not conically independent).