v2009.01.01 - Convex Optimization

310 CHAPTER 4. SEMIDEFINITE PROGRAMMING
4.6.0.0.2 Example. Procrustes problem. [45]
Example 4.6.0.0.1 is extensible. An orthonormal matrix Q∈ R n×p is
completely characterized by Q T Q = I . Consider the particular case
Q = [x y ]∈ R n×2 as variable to a Procrustes problem (C.3): given
A∈ R m×n and B ∈ R m×2
minimize ‖AQ − B‖ F
Q∈R n×2
subject to Q T Q = I
(704)
which is nonconvex. By vectorizing matrix Q we can make the assignment:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x [ x T y T 1] X Z x xx T xy T x
G = ⎣ y ⎦ = ⎣ Z T Y y ⎦=
∆ ⎣ yx T yy T y ⎦∈ S 2n+1 (705)
1
x T y T 1 x T y T 1
Now orthonormal Procrustes problem (704) can be equivalently restated:
minimize
X , Y , Z , x , y
‖A[x y ] − B‖ F
⎡
X Z
⎤
x
subject to G = ⎣ Z T Y y ⎦
x T y T 1
(G ≽ 0)
rankG = 1
trX = 1
trY = 1
trZ = 0
(706)
To solve this, we form the convex problem sequence:
minimize
X , Y , Z , x , y
‖A[x y ] −B‖ F + 〈G, W 〉
⎡
X Z
⎤
x
subject to G = ⎣ Z T Y y ⎦ ≽ 0
x T y T 1
trX = 1
trY = 1
trZ = 0
(707)