v2007.09.17 - Convex Optimization

4.4. RANK-CONSTRAINED SEMIDEFINITE PROGRAM 277in symmetric variable matrix X ∈ S 2 and variable vector z ∈ R 2 where⎡0 − 1 12A = ⎣ − 1 0 021 0 3⎤⎦ (659)Then the method of convex iteration from4.4.1 is applied to implement therank constraint.4.4.3.0.4 Example. Procrustes problem. [38]Example 4.4.3.0.2 is extensible. An orthonormal matrix Q∈ R n×p iscompletely characterized by Q T Q = I . Consider the particular caseQ = [x y ]∈ R n×2 as variable to a Procrustes problem (C.3): givenA∈ R m×n and B ∈ R m×2minimize ‖AQ − B‖ FQ∈R n×2subject to Q T Q = I(660)which is nonconvex. By vectorizing matrix Q we can make the assignment:⎡ ⎤x [x T y T 1]⎡G = ⎣ y ⎦ = ⎣1X Z xZ T Y yx T y T 1⎤⎡⎦=∆ ⎣xx T xy T xyx T yy T yx T y T 1Now Procrustes problem (660) can be equivalently restated:⎤⎦∈ S 2n+1 (661)minimizeX , Y , Z , x , y‖A[x y ] − B‖ F⎡X Z⎤xsubject to G = ⎣ Z T Y y ⎦x T y T 1(G ≽ 0)rankG = 1trX = 1trY = 1trZ = 0(662)