v2009.01.01 - Convex Optimization

266 CHAPTER 4. SEMIDEFINITE PROGRAMMING
4.2.3.1.2 Example. Optimization on elliptope versus 1-norm polyhedron
for minimum cardinality Boolean Example 4.2.3.1.1.
A minimum cardinality problem is typically formulated via, what is by now,
the standard practice [101] [59,3.2,3.4] of column normalization applied to
a 1-norm problem surrogate like (460). Suppose we define a diagonal matrix
⎡
Λ =
∆ ⎢
⎣
⎤
‖A(:,1)‖ 2 0
‖A(:, 2)‖ 2 ⎥
...
0 ‖A(:, 6)‖ 2
⎦ ∈ S6 (630)
used to normalize the columns (assumed nonzero) of given noiseless data
matrix A . Then approximate the minimum cardinality Boolean problem
as
where optimal solution
minimize ‖x‖ 0
x
subject to Ax = b
x i ∈ {0, 1} ,
i=1... n
minimize ‖ỹ‖ 1
ỹ
subject to AΛ −1 ỹ = b
1 ≽ Λ −1 ỹ ≽ 0
(614)
(631)
y ⋆ = round(Λ −1 ỹ ⋆ ) (632)
The inequality in (631) relaxes Boolean constraint y i ∈ {0, 1} from (614);
bounding any solution y ⋆ to a nonnegative unit hypercube whose vertices are
binary numbers. Convex problem (631) is justified by the convex envelope
cenv ‖x‖ 0 on {x∈ R n | ‖x‖ ∞ ≤κ} = 1 κ ‖x‖ 1 (1270)
Donoho concurs with this particular formulation, equivalently expressible as
a linear program via (456).
Approximation (631) is therefore equivalent to minimization of an affine
function on a bounded polyhedron, whereas semidefinite program
minimize 1 Tˆx
X∈ S n , ˆx∈R n
subject to A(ˆx + 1) 1
[ = b 2
] X ˆx
G =
ˆx T 1
δ(X) = 1
≽ 0
(626)