v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 317
〈G ⋆ , Y 〉). To overcome this behavior, we introduce a heuristic into the
implementation on Wıκımization that momentarily reverses direction of
search (like (701)) upon stall detection. We find that rate of convergence
can be sped significantly by detecting stalls early.
4.6.0.0.7 Example. Variable-vector normalization.
Suppose, within some convex optimization problem, we want vector variables
x, y ∈ R N constrained by a nonconvex equality:
x‖y‖ = y (723)
id est, ‖x‖ = 1 and x points in the same direction as y≠0 ; e.g.,
minimize f(x, y)
x , y
subject to (x, y) ∈ C
x‖y‖ = y
(724)
where f is some convex function and C is some convex set. We can realize
the nonconvex equality by constraining rank and adding a regularization
term to the objective. Make the assignment:
⎡ ⎤ ⎡
x [x T y T 1]
G = ⎣ y ⎦ = ⎣
1
X Z x
Z Y y
x T y T 1
⎤
⎡
⎦=
∆ ⎣
xx T xy T x
yx T yy T y
x T y T 1
⎤
⎦∈ S 2N+1 (725)
where X , Y ∈ S N , also Z ∈ S N [sic] . Any rank-1 solution must take the
form of (725). (B.1) The problem statement equivalent to (724) is then
written
minimize
X , Y , Z , x , y
subject to (x, y) ∈ C
⎡
f(x, y) + ‖X − Y ‖ F
G = ⎣
rankG = 1
(G ≽ 0)
tr(X) = 1
δ(Z) ≽ 0
X Z x
Z Y y
x T y T 1
⎤
⎦
(726)