v2009.01.01 - Convex Optimization

308 CHAPTER 4. SEMIDEFINITE PROGRAMMING
where x ⋆ = [I −I ]a ⋆ ; from which it may be construed that any vector
1-norm minimization problem has equivalent expression in a nonnegative
variable.
4.6 Cardinality and rank constraint examples
4.6.0.0.1 Example. Projection on an ellipsoid boundary. [45] [123,5.1]
[213,2] Consider classical linear equation Ax = b but with a constraint on
norm of solution x , given matrices C , fat A , and vector b∈R(A)
find x ∈ R N
subject to Ax = b
‖Cx‖ = 1
(696)
The set {x | ‖Cx‖=1} describes an ellipsoid boundary (Figure 12). This
is a nonconvex problem because solution is constrained to that boundary.
Assign
G =
[ Cx
1
] [ ] [ [x T C T 1] X Cx ∆ Cxx =
x T C T =
T C T Cx
1 x T C T 1
]
∈ S N+1
(697)
Any rank-1 solution must have this form. (B.1.0.2) Ellipsoidally constrained
feasibility problem (696) is equivalent to:
find
X∈S N
x ∈ R N
subject to Ax = b
[
X Cx
G =
x T C T 1
(G ≽ 0)
rankG = 1
trX = 1
]
(698)
This is transformed to an equivalent convex problem by moving the rank
constraint to the objective: We iterate solution of