v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 207
rather
x > 0, y ≥ 1 x
⇔
[ x 1
1 y
]
≽ 0 (478)
(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
≡
minimize
x∈R n , y∈R
subject to
y
[
√ xi
n
√ ] n
≽ 0 , y
i=1... n
x ∈ C (479)
x ≻ 0, y ≥ tr ( δ(x) −1)
⇔
[
xi
√ n
√ n y
]
≽ 0 , i=1... n (480)
3.1.5 fractional power
[124] 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 (481)
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
3.1.5.1 positive
Then we have the equivalent semidefinite program for maximizing a concave
function x α , for quantized 0≤α 0
x ∈ C
≡
maximize
x∈R , y∈R q+1
subject to
y q
[
yi−1 y i
y i x b i
]
≽ 0 , i=1... q
x ∈ C (482)