v2009.01.01 - Convex Optimization

206 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.1.3.2 clipping
Clipping negative vector entries is accomplished:
where, for x = [x i ]∈ R n
x + = t ⋆ =
‖x + ‖ 1 = minimize 1 T t
t∈R n
subject to x ≼ t
[
xi , x i ≥ 0
0, x i < 0
}
0 ≼ t
, i=1... n
]
(472)
(473)
(clipping)
minimize
x∈R n ‖x + ‖ 1
subject to x ∈ C
≡
minimize 1 T t
x∈R n , t∈R n
subject to x ≼ t
0 ≼ t
x ∈ C
(474)
3.1.4 inverted
We wish to implement objectives of the form x −1 . Suppose we have a 2×2
matrix
[ ]
T =
∆ x z
∈ R 2 (475)
z y
which is positive semidefinite by (1413) when
T ≽ 0 ⇔ x > 0 and xy ≥ z 2 (476)
A polynomial constraint such as this is therefore called a conic constraint. 3.9
This means we may formulate convex problems, having inverted variables,
as semidefinite programs in Schur-form; e.g.,
minimize x −1
x∈R
subject to x > 0
x ∈ C
≡
minimize
x , y ∈ R
subject to
y
[ ] x 1
≽ 0
1 y
x ∈ C
(477)
3.9 In this dimension the cone is called a rotated quadratic or circular cone, or positive
semidefinite cone.