l1-magic : Recovery of Sparse Signals via Convex Programming

Appendix
A
l 1 minimization with equality constraints
When x, A and b are real, then (P 1 ) can be recast as the linear program
∑
min u i subject to x i − u i ≤ 0
x,u
i
−x i − u i ≤ 0,
Ax = b
which can be solved using the standard primal-dual algorithm outlined in Section 2.1 (again,
see [2, Chap.11] for a full discussion). Set
f u1;i := x i − u i
f u2;i := −x i − u i ,
with λ u1;i, λ u2;i the corresponding dual variables, and let f u1 be the vector (f u1;1 . . . f u1;N ) T
(and likewise for f u2 , λ u1 , λ u2 ). Note that
( )
δi
∇f u1;i = ,
−δ i
( ) −δi
∇f u2;i = ,
−δ i
∇ 2 f u1;i = 0, ∇ 2 f u2;i = 0,
where δ i is the standard basis vector for component i. Thus at a point (x, u; v, λ u1 , λ u2 ), the
central and dual residuals are
( ) −Λu1 f
r cent =
u1
− (1/τ)1,
−Λ u2 f u2
( )
λu1 − λ
r dual =
u2 + A T v
,
1 − λ u1 − λ u2
and the Newton step (5) is given by:
⎛
⎝ Σ ⎞ ⎛
1 Σ 2 A T
Σ 2 Σ 1 0 ⎠ ⎝ ∆x ⎞ ⎛
∆u⎠ = ⎝ w ⎞ ⎛
⎞
−1
1 (−1/τ) · (−fu 1
+ fu −1
2
) − A T v
w 2
⎠ := ⎝ −1 − (1/τ) · (fu −1
1
+ fu −1
2
) ⎠ ,
A 0 0 ∆v w 3 b − Ax
with
Σ 1 = Λ u1 F −1
u 1
− Λ u2 Fu −1
2
, Σ 2 = Λ u1 Fu −1
1
+ Λ u2 Fu −1
2
,
(The F • , for example, are diagonal matrices with (F • ) ii = f •;i , and f −1
•;i
we can eliminate
and solve
Σ x = Σ 1 − Σ 2 2Σ −1
1 ,
∆x = Σ −1
x (w 1 − Σ 2 Σ −1
1 w 2 − A T ∆v)
∆u = Σ −1
1 (w 2 − Σ 2 ∆x),
−AΣ −1
1 AT ∆v = w 3 − A(Σ −1
x w 1 − Σ −1
x Σ 2 Σ −1
1 w 2).
= 1/f •;i.) Setting
This is a K×K positive definite system of equations, and can be solved using conjugate gradients.
Given ∆x, ∆u, ∆v, we calculate the change in the inequality dual variables as in (4):
∆λ u1 = Λ u1 Fu −1
1
(−∆x + ∆u) − λ u1 − (1/τ)fu −1
1
∆λ u2 = Λ u2 Fu −1
2
(∆x + ∆u) − λ u2 − (1/τ)fu −1
2
.
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