v2009.01.01 - Convex Optimization

4.2. FRAMEWORK 263
The pseudoinverse solution (rounded)
⎡ ⎤
−0.0456
−0.1881
x P
= A † b =
0.0623
⎢ 0.2668
⎥
⎣ 0.3770 ⎦
−0.1102
(619)
has least norm ‖x P
‖ 2 =0.5165 ; id est, the optimal solution to (E.0.1.0.1)
minimize ‖x‖ 2
x
subject to Ax = b
(620)
Certainly none of the traditional methods provide x ⋆ = e 4 (617) because, and
in general, for Ax = b
∥
∥arg inf ‖x‖ 2
∥
∥2 ≤ ∥ ∥arg inf ‖x‖ 1
∥
∥2 ≤ ∥ ∥arg inf ‖x‖ 0
∥
∥2 (621)
We can reformulate this minimum cardinality Boolean problem (614) as
a semidefinite program: First transform the variable
x ∆ = (ˆx + 1) 1 2
(622)
so ˆx i ∈ {−1, 1} ; equivalently,
minimize ‖(ˆx + 1) 1‖ ˆx
2 0
subject to A(ˆx + 1) 1 = b 2
δ(ˆxˆx T ) = 1
(623)
where δ is the main-diagonal linear operator (A.1). By assigning (B.1)
[ ˆx
G =
1
] [ ˆx T 1] =
[ X ˆx
ˆx T 1
]
[
∆ ˆxˆxT ˆx
=
ˆx T 1
]
∈ S n+1 (624)