v2009.01.01 - Convex Optimization

534 CHAPTER 7. PROXIMITY PROBLEMS
verifiable by observing conic dependencies (2.10.3) among the aggregate of
halfspace-description normals. The cone membership constraint in (1311a)
therefore inherently insures existence of a symmetric hollow matrix.
Optimization (1311b) would be a Procrustes problem (C.4) were it
not for the hollowness constraint; it is, instead, a minimization over the
intersection of the nonconvex manifold of orthogonal matrices with another
nonconvex set in variable R specified by the hollowness constraint.
We solve problem (1311b) by a method introduced in4.6.0.0.2: Define
R = [r 1 · · · r N+1 ]∈ R N+1×N+1 and make the assignment
⎡
G = ⎢
⎣
r 1
.
r N+1
1
⎤
[ r1 T · · · rN+1 T 1] ⎥
∈ S (N+1)2 +1
⎦
⎡
⎤ ⎡
R 11 · · · R 1,N+1 r 1
.
...
.
∆
= ⎢
⎣ R1,N+1 T ⎥=
⎢
R N+1,N+1 r N+1 ⎦ ⎣
r1 T · · · rN+1 T 1
(1314)
⎤
r 1 r1 T · · · r 1 rN+1 T r 1
.
...
.
r N+1 r1 T r N+1 rN+1 T ⎥
r N+1 ⎦
r1 T · · · rN+1 T 1
where R ∆ ij = r i rj T ∈ R N+1×N+1 and Υ ⋆ ii ∈ R . Since R Υ ⋆ R T = N+1 ∑
ΥiiR ⋆ ii then
problem (1311b) is equivalently expressed:
∥ ∥∥∥ N+1
2
∑
minimize Υ ⋆
R ij , r i
iiR ii − Λ
∥
i=1
F
subject to trR ii = 1, i=1... N+1
trR ij ⎡
= 0,
⎤
i