Chapter 3 Geometry of convex functions - Meboo Publishing ...

3.4. INVERTED FUNCTIONS AND ROOTS 227
which is positive semidefinite by (1487) when
T ≽ 0 ⇔ x > 0 and xy ≥ z 2 (516)
A polynomial constraint such as this is therefore called a conic constraint. 3.12
This means we may formulate convex problems, having inverted variables,
as semidefinite programs in Schur-form; e.g.,
rather
minimize x −1
x∈R
subject to x > 0
x ∈ C
≡
x > 0, y ≥ 1 x
⇔
minimize
x , y ∈ R
subject to
[ x 1
1 y
(inverted) For vector x=[x i , i=1... n]∈ R n
minimize
x∈R n
n∑
i=1
x −1
i
subject to x ≻ 0
rather
x ∈ C
≡
x ≻ 0, y ≥ tr ( δ(x) −1)
minimize
x∈R n , y∈R
subject to
⇔
y
[ ] x 1
≽ 0
1 y
x ∈ C
(517)
]
≽ 0 (518)
y
[
√ xi
n
√ ] n
≽ 0 , y
i=1... n
x ∈ C (519)
[
xi
√ n
√ n y
]
≽ 0 , i=1... n (520)
3.4.1 fractional power
[145] To implement an objective of the form x α for positive α , we quantize
α and work instead with that approximation. Choose nonnegative integer q
for adequate quantization of α like so:
α k 2 q (521)
where k ∈{0, 1, 2... 2 q −1}. Any k from that set may be written
∑
k= q b i 2 i−1 where b i ∈ {0, 1}. Define vector y=[y i , i=0... q] with y 0 =1:
i=1
3.12 In this dimension, the convex cone formed from the set of all values {x , y , z} satisfying
constraint (516) is called a rotated quadratic or circular cone or positive semidefinite cone.