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

228 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.4.1.1 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 (522)
where nonnegativity of y q is enforced by maximization; id est,
x > 0, y q ≤ x α
⇔
[
yi−1 y i
y i
x b i
]
≽ 0 , i=1... q (523)
3.4.1.2 negative
It is also desirable to 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 (524)
]
[ ]
z 1
≽ 0
1 y q
≽ 0 ,
i=1... q
(525)