v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 309
with
minimize 〈G , Y 〉
X∈S N , x∈R N
subject to Ax = b
[
X Cx
G =
x T C T 1
trX = 1
minimize
Y ∈ S N+1 〈G ⋆ , Y 〉
subject to 0 ≼ Y ≼ I
trY = N
]
≽ 0
(699)
(700)
until convergence. Direction matrix Y ∈ S N+1 , initially 0, controls rank.
(1581a) Singular value decomposition G ⋆ = UΣQ T ∈ S N+1
+ (A.6) provides a
new direction matrix Y = U(:, 2:N+1)U(:, 2:N+1) T that optimally solves
(700) at each iteration. An optimal solution to (696) is thereby found in a
few iterations, making convex problem (699) its equivalent.
It remains possible for the iteration to stall; were a rank-1 G matrix not
found. In that case, the current search direction is momentarily reversed
with an added random element:
Y = −U(:,2:N+1) ∗ (U(:,2:N+1) ′ + randn(N,1) ∗U(:,1) ′ ) (701)
in Matlab notation. This heuristic is quite effective for problem (696) which
is exceptionally easy to solve by convex iteration.
When b /∈R(A) then problem (696) must be restated as a projection:
minimize ‖Ax − b‖
x∈R N
subject to ‖Cx‖ = 1
(702)
This is a projection of point b on an ellipsoid boundary because any affine
transformation of an ellipsoid remains an ellipsoid. Problem (699) in turn
becomes
minimize 〈G , Y 〉 + ‖Ax − b‖
X∈S N , x∈R N
[ ]
X Cx
subject to G =
x T C T ≽ 0 (703)
1
trX = 1
We iterate this with calculation (700) of direction matrix Y as before until
a rank-1 G matrix is found.