v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 315
4.6.0.0.5 Example. Tractable polynomial constraint.
The ability to handle rank constraints makes polynomial constraints
(generally nonconvex) transformable to convex constraints. All optimization
problems having polynomial objective and polynomial constraints can be
reformulated as a semidefinite program with a rank-1 constraint. [247]
Suppose we require
3 + 2x − xy ≤ 0 (715)
Assign
⎡ ⎤
x [x y 1]
G = ⎣ y ⎦ =
1
[ X z
z T 1
]
∆
=
⎡ ⎤
x 2 xy x
⎣ xy y 2 y ⎦∈ S 3 (716)
x y 1
The polynomial constraint (715) is equivalent to the constraint set (B.1.0.2)
tr(GA) ≤ 0
[ X z
G =
z T 1
rankG = 1
]
(≽ 0)
(717)
in symmetric variable matrix X ∈ S 2 and variable vector z ∈ R 2 where
⎡ ⎤
0 − 1 1
2
A = ⎣ − 1 0 0 ⎦ (718)
2
1 0 3
Then the method of convex iteration from4.4.1 is applied to implement the
rank constraint.
4.6.0.0.6 Example. Boolean vector feasible to Ax ≼ b. (confer4.2.3.1.1)
Now we consider solution to a discrete problem whose only known analytical
method of solution is combinatorial in complexity: given A∈ R M×N and
b∈ R M find x ∈ R N
subject to Ax ≼ b
δ(xx T ) = 1
(719)
This nonconvex problem demands a Boolean solution [x i = ±1, i=1... N ].