Detailed Derivation of the Semidefinite Programming
Optimization Objective Function for Equation 4
Hui Wu∗1 , Yi Fang2 , Huming Wu3 , and Shenhong Zhu3
1
2
Google, 1600 Amphitheatre Parkway, Mountain View, California 94043, USA
Santa Clara University, 500 El Camino Real, Santa Clara, California 95053, USA
3
Yahoo!, 701 1st Ave, Sunnyvale, California 94089, USA
steps
arg max
S⊂E
|S|=k
T =E\S
1
|S|×|T |
P
1
− |S|×|S|
s∈S
t∈T
w(s, t)
P
si ∈S
sj ∈S
w(si , sj ))
P P 2+α
arg min
( α xi xj + (xi + xj ))wij
2+α
⇓ × α ,P
as 2+α
> 0, the solution will not be changed
α
P
)2 xi xj + 2+α
(xi + xj ))wij
= arg min
(( 2+α
α
α
⇓ + P P wP
,
which
is
a
constant, the solution will be unchanged
ij P
= arg min
(( 2+α
)2 xi xj + 2+α
(xi + xj ) + 1)wij
α
α
(1)
Firstly, let x be the vector indicating the assignments of this
partition, and
xi =
1
qi is assigned to S
−1 qi is assigned to T .
arg min
PP
xi + 1)( 2+α
xj + 1)wij
( 2+α
α
α
(7)
(2)
We furthermore assume |T | = α|S| and let wij represent
w(qi , qj ), therefore Equation 1 is reformulated as
PP
arg max(
P(4
P− 2α)wij
−2
((2 + α)xi xj + α(xi + xj ))wij ),
=
By combining Equation 6 and 7, we have
arg min
(3)
s.t.
( 2+α
xi + 1)( 2+α
xj + 1)wij
α
α
// The following one is driven from
2+α
xi
α
whose details can be found at Appendix .
PP
+ 1 ∈ {2 +
2 −2
, }
α α
xi ∈ {−1, 1}
//The following two are driven
// from
P
k) = 2k − n
P 2+α i xi = 1 ∗2 k + (−1) ∗ 2(n −
)2 ∗ (n − k)
( α xi + 1) = (2 + α )2 ∗ k + ( −2
α
As
(4 − 2α)wij is a constant, then Equation 3 is simplied as
PP
P 2+α
( α xi + 1) = (2 +
2
)
α
∗ k + ( −2
) ∗ (n − k)
α
(8)
PP
arg max −2 ×
((2 + α)xi xj + α(xi + xj ))wij
⇓ ÷2, which does P
not change the solution
P
= arg max −1 ×
((2 + α)xi xj + α(xi + xj ))wij
⇓ changingP"arg
Pmax(−1)" to "arg min" does not change the solution
= arg min
((2 + α)xi xj + α(xi + xj ))wij
(4)
The further simplied form is
XX 2 + α
arg min
(
xi xj + (xi + xj ))wij
α
Let
yi =
s.t.
PP
arg min
(5)
s.t.
P
i
xi = 1 ∗ k + (−1) ∗ (n − k) = 2k − n
We continue reformulating Equation 6 with the following
wij yi yj
2 2
)
α
P
yi2 = (2 +
2 2
)
α
P
yi = (2 +
2
)
α
i
(6)
PP
yi2 − 2yi = (1 +
i
( 2+α
xi xj + (xi + xj ))wij
α
xi ∈ {−1, 1}
(9)
and then Equation 8 is reformulated as
By adding the constraint of xi ∈ {−1, 1} and |S| = k to
Equation 5, and then we have the following system
arg min
2+α
xi + 1,
α
−1
∗ k + ( −2
)2 ∗ (n − k))
α
(10)
∗ k + ( −2
) ∗ (n − k)
α
whose details are described in Appendix .
we further bring another two variables z, and X , where
z = (y1 , y2 , · · · , yn , 1n+1 , 1n+2 , · · · , 12n )T ,
(11)
X = zzT .
(12)
and
Under the current context, we get the following system:
where
C=
arg minhC, XiS 2n
s.t. ∀1 ≤ i ≤ n, hAi , XiS 2n = (1 +
2 2
)
α
2 2
)
α
hA2n+2 , XiS 2n = (2 +
2
)
α
z = (y1 , y2 , · · · , yn , 1n+1 , 1n+2 , · · · , 12n )T
∗ k + ( −2
)2 ∗ (n − k)
α
yi =
∗ k + ( −2
) ∗ (n − k)
α
where
Wn×n 0n×n
0n×n 0n×n
n+k
x
n−k i
+1
∀1 ≤ i ≤
2n − 1, Ai ∈ R2n×2n , where
a = b = i;
 1


−1 a = i, b = 2n;
Ai,a,b =
−1 a = 2n, b = i;


 0
others.
(13)
X = zzT
X0
C=
Wn×n 0n×n
0n×n 0n×n
where W's elements are dened in Equation ??.
−1
∀n + 1 ≤ i ≤ 2n, hAi , XiS 2n = −1
hA2n+1 , XiS 2n = (2 +
A2n ∈ R2n×2n
 , where
a = b = 2n;
 1


−1 a = 2n − 1, b = 2n;
A2n,a,b =
−1 a = 2n, b = 2n − 1;


 0
others.
∀1 ≤ i ≤
2n − 1, Ai ∈ R2n×2n , where
1
a = b = i;



−1 a = i, b = 2n;
Ai,a,b =
−1 a = 2n, b = i;


 0
others.
A2n+1 ∈ R2n×2n
, where
1 a = b, 1 ≤ a, b ≤ n;
A2n+1,a,b =
0 others.
A2n ∈ R2n×2n
 , where
a = b = 2n;
 1


−1 a = 2n − 1, b = 2n;
A2n,a,b =
−1 a = 2n, b = 2n − 1;


 0
others.
A2n+2 ∈ R2n×2n
 , where
 0.5 1 ≤ a ≤ n, b = 2n;
0.5 a = 2n, 1 ≤ b ≤ n;
A2n+2,a,b =
 0
others.
Formulation A
A2n+1 ∈ R , where
1 a = b, 1 ≤ a, b ≤ n;
A2n+1,a,b =
0 others.
P
1
arg max( |S|∗|T
qi ∈S,qj ∈T wij
|
P
1
− |S|∗|S|
qi ∈S,qj ∈S wij )
2n×2n
As |T | = α|S|, then
A2n+2 ∈ R2n×2n
 , where
 0.5 1 ≤ a ≤ n, b = 2n;
0.5 a = 2n, 1 ≤ b ≤ n;
A2n+2,a,b =
 0
others.
=
P P 1−xi xj
1
arg max( |S|∗α|S|
wij
P P (xi +xj )(2+xi2+xj )
1
− |S|∗|S|
wij )
8
=
P P 1−xi xj
1
1
arg max |S|∗|S|
(
wij
2
P P (xi +xj )∗(2+xi +xj ) α
−
w
)
ij
4
=
PP
1
1
arg max |S|∗|S|
∗ 12 ∗ 4∗α
(4 ∗ (1 − xi xj )
−α(xi + xj )(2 + xi + xj ))wij
The reformulation details are given by Appendix . After
replacing α by n−k
, the system is further reformulated as:
k
arg min tr(C T X)
s.t. ∀1 ≤ i ≤ n, tr(ATi X) = (1 +
2×k 2
)
n−k
As
−1
∀n + 1 ≤ i ≤ 2n, tr(ATi X) = −1
tr(AT2n+1 X) = (2 +
2×k 2
)
n−k
tr(AT2n+2 X) = (2 +
2×k
)
n−k
∗ k + ( −2×k
)2 ∗ (n − k)
n−k
∗ k + ( −2×k
) ∗ (n − k)
n−k
X0
(14)
1
|S|∗|S|
∗
1
2
∗
1
4∗α
is constant.
PP
=
arg max
(4 ∗ (1 − xi xj )
−α(xi + xj )(2 + xi + xj ))wij
=
PP
arg max
(4 − 4xi xj − 2αxi − αx2i
−αxi xj − 2αxj − αxi xj − αx2j )wij
As xi ∈ {−1, 1}, then x2i = 1
=
PP
arg max
(4 − 4xi xj
−2αxi − α − αxi xj − 2αxj − αxi xj − α)wij
By using X = zzT , the above system is reformulated as
=
PP
arg max
((4 − 2α)
−((4 + 2α)xi xj + 2αxi + 2αxj ))wij
arg minhC, XiS 2n
s.t. ∀1 ≤ i ≤ n, hAi , XiS 2n = (1 +
//From cstr. 1
PP
= arg P
max(
(4 − 2α)wij
P
−2
((2 + α)xi xj + α(xi + xj ))wij )
Formulation B
and then Equation 8 can be reformulated as
arg min
s.t.
yi ∈ {2 +
2 2
)
α
P
yi = (2 +
2
)
α
hA2n+2 , XiS 2n = (2 +
s.t.
PP
∗ k + ( −2
) ∗ (n − k))
α
wij yi yj
yi2 − 2yi = (1 +
2 2
)
α
P
yi2 = (2 +
2 2
)
α
P
yi = (2 +
2
)
α
i
i
−1
∗ k + ( −2
)2 ∗ (n − k))
α
∗ k + ( −2
) ∗ (n − k)
α
Formulation C
arg min zT C z
where
Wn×n 0n×n
0n×n 0n×n
2 2
)
α
2
)
α
zi2 − 2 ∗ zi = −1
∗ k + ( −2
)2 ∗ (n − k)
α
∗ k + ( −2
) ∗ (n − k)
α
X0
where
C=
Wn×n 0n×n
0n×n 0n×n
∀1 ≤ i ≤
2n − 1, Ai ∈ R2n×2n , where
1
a = b = i;



−1 a = i, b = 2n;
Ai,a,b =
−1 a = 2n, b = i;


 0
others.
This objective function has the following constraints:
1. ∀1 ≤ i ≤ n, zT Ai z = (1 + α2 )2 − 1, where
Ai,a,b
then
A2n ∈ R2n×2n
 , where
1
a = b = 2n;



−1 a = 2n − 1, b = 2n;
A2n,a,b =
 −1 a = 2n, b = 2n − 1;

 0
others.
The objective function is
C=
//From cstr. 4
∗ k + ( −2
)2 ∗ (n − k))
α
The system is further re-formulated as
arg min
//From cstr. 3
2 −2
, }
α α
yi2 = (2 +
(zi − 1)2 = 0,
hA2n+1 , XiS 2n = (2 +
wij yi yj
P

1



−1
=
 −1

 0
a = b = i;
a = i, b = 2n;
a = 2n, b = i;
others.
2. ∀n + 1 ≤ i ≤ 2n, zi = 1
3. (2 + α2 )2 ∗ k + ( −2
)2 ∗ (n − k))
α
T
= z A2n+1 z , where
A2n+1 ∈ R2n×2n
,
1 a = b, 1 ≤ a, b ≤ n;
A2n+1,a,b =
0 others.
4. (2 + α2 ) ∗ k + ( −2
) ∗ (n − k)
α
= zT A2n+2 z, where
A2n+2 ∈ R2n×2n
 ,
 0.5 1 ≤ a ≤ n, b = 2n;
0.5 a = 2n, 1 ≤ b ≤ n;
A2n+2,a,b =
 0
others.
−1
∀n + 1 ≤ i ≤ 2n, hAi , XiS 2n = −1
//From cstr. 2, as
PP
2 2
)
α
A2n+1 ∈ R2n×2n
, where
1 a = b, 1 ≤ a, b ≤ n;
A2n+1,a,b =
0 others.
A2n+2 ∈ R2n×2n
 , where
 0.5 1 ≤ a ≤ n, b = 2n;
0.5 a = 2n, 1 ≤ b ≤ n;
A2n+2,a,b =
 0
others.