v2009.01.01 - Convex Optimization

208 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
where nonnegativity of y q is enforced by maximization; id est,
[ ]
x > 0, y q ≤ x α yi−1 y i
⇔
≽ 0 , i=1... q (483)
y i
x b i
3.1.5.2 negative
Is it also desirable implement an objective of the form x −α for positive α .
The technique is nearly the same as before: for quantized 0≤α 0
rather
x ∈ C
x > 0, z ≥ x −α
≡
⇔
minimize
x , z∈R , y∈R q+1
subject to
[
yi−1 y i
y i
x b i
z
[
yi−1 y i
y i
x b i
]
[ ]
z 1
≽ 0
1 y q
≽ 0 ,
i=1... q
x ∈ C (484)
]
[ ]
z 1
≽ 0
1 y q
≽ 0 ,
i=1... q
(485)
3.1.5.3 positive inverted
Now define vector t=[t i , i=0... q] with t 0 =1. To implement an objective
x 1/α for quantized 0≤α