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TEACHING
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Commits
dcbab056
Commit
dcbab056
authored
Jul 27, 2016
by
Gianluca Frison
Browse files
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added solution to ex06
parent
0dc685c5
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7 changed files
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480 additions
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+480
-0
cows.mod
exercises/ex06/solutions/cows.mod
+31
-0
cows_robust.mod
exercises/ex06/solutions/cows_robust.mod
+38
-0
cows_worst.mod
exercises/ex06/solutions/cows_worst.mod
+35
-0
nug12.m
exercises/ex06/solutions/nug12.m
+83
-0
nug30.m
exercises/ex06/solutions/nug30.m
+149
-0
sol06.txt
exercises/ex06/solutions/sol06.txt
+26
-0
tai30a.m
exercises/ex06/solutions/tai30a.m
+118
-0
No files found.
exercises/ex06/solutions/cows.mod
0 → 100644
View file @
dcbab056
reset;
param b{1..2};
param a{1..2,1..2};
var x{1..2} >= 0;
maximize obj: 250*x[1] + 45*x[2];
s.t. bd1: x[1] <= 50;
s.t. bd2: x[1] <= 300;
s.t. ineq{i in 1..2}: sum{j in 1..2} a[i,j]*x[j] <= b[i];
data;
param b :=
1 72
2 10000;
param a: 1 2:=
1 1 0.2
2 150 25;
option solver ipopt;
solve;
display _obj;
display _var;
display sum{j in 1..2} a[1,j]*x[j] - b[1];
display sum{j in 1..2} a[2,j]*x[j] - b[2];
exercises/ex06/solutions/cows_robust.mod
0 → 100644
View file @
dcbab056
reset;
param b{1..2};
param a{1..2,1..2};
param ahat{i in 1..2,j in 1..2} := a[i,j]*0.05;
param omega = 0.1;
var x{1..2} >= 0 := 1;
var y{1..2} >= 0 := 1;
var z{1..2} >= 0 := 1;
maximize obj: 250*x[1] + 45*x[2];
s.t. bd1: x[1] <= 50;
s.t. bd2: x[2] <= 300;
s.t. neq{i in 1..2}: sum{j in 1..2} (a[i,j]*x[j] + ahat[i,j]*y[j])
+ omega*sqrt(sum{j in 1..2} ahat[i,j]^2*z[j]^2) <= b[i];
s.t. bd3{i in 1..2}: - y[i] <= x[i] - z[i];
s.t. bd4{i in 1..2}: x[i] - z[i] <= y[i];
data;
param b :=
1 72
2 10000;
param a: 1 2 :=
1 1 0.2
2 150 25;
option solver ipopt;
solve;
display _obj;
display _var;
display sum{j in 1..2} a[1,j]*x[j] - b[1];
display sum{j in 1..2} a[2,j]*x[j] - b[2];
exercises/ex06/solutions/cows_worst.mod
0 → 100644
View file @
dcbab056
reset;
param b{1..2};
param a{1..2,1..2};
param ahat{i in 1..2,j in 1..2} := a[i,j]*0.05;
var x{1..2} >= 0;
var y{1..2} >= 0;
maximize obj: 250*x[1] + 45*x[2];
s.t. bd1: x[1] <= 50;
s.t. bd2: x[2] <= 300;
s.t. neq{i in 1..2}: sum{j in 1..2} (a[i,j]*x[j] + ahat[i,j]*y[j]) <= b[i];
s.t. bd3{i in 1..2}: - y[i] <= x[i];
s.t. bd4{i in 1..2}: x[i] <= y[i];
data;
param b :=
1 72
2 10000;
param a: 1 2 :=
1 1 0.2
2 150 25;
option solver ipopt;
solve;
display _obj;
display _var;
display sum{j in 1..2} a[1,j]*x[j] - b[1];
display sum{j in 1..2} a[2,j]*x[j] - b[2];
exercises/ex06/solutions/nug12.m
0 → 100644
View file @
dcbab056
B
=
[
0
1
2
3
1
2
3
4
2
3
4
5
;
1
0
1
2
2
1
2
3
3
2
3
4
;
2
1
0
1
3
2
1
2
4
3
2
3
;
3
2
1
0
4
3
2
1
5
4
3
2
;
1
2
3
4
0
1
2
3
1
2
3
4
;
2
1
2
3
1
0
1
2
2
1
2
3
;
3
2
1
2
2
1
0
1
3
2
1
2
;
4
3
2
1
3
2
1
0
4
3
2
1
;
2
3
4
5
1
2
3
4
0
1
2
3
;
3
2
3
4
2
1
2
3
1
0
1
2
;
4
3
2
3
3
2
1
2
2
1
0
1
;
5
4
3
2
4
3
2
1
3
2
1
0
];
A
=
[
0
5
2
4
1
0
0
6
2
1
1
1
;
5
0
3
0
2
2
2
0
4
5
0
0
;
2
3
0
0
0
0
0
5
5
2
2
2
;
4
0
0
0
5
2
2
10
0
0
5
5
;
1
2
0
5
0
10
0
0
0
5
1
1
;
0
2
0
2
10
0
5
1
1
5
4
0
;
0
2
0
2
0
5
0
10
5
2
3
3
;
6
0
5
10
0
1
10
0
0
0
5
0
;
2
4
5
0
0
1
5
0
0
0
10
10
;
1
5
2
0
5
5
2
0
0
0
5
0
;
1
0
2
5
1
4
3
5
10
5
0
2
;
1
0
2
5
1
0
3
0
10
0
2
0
];
% swap A and B
%C=A; A=B; B=C;
n
=
12
;
n2
=
2
*
n
;
e
=
ones
(
n
,
1
);
E
=
e
*
e
'
;
I
=
eye
(
n
);
% split B = Bp - Bm, with Bp and Bm positive semidefinite
[
V
,
D
]
=
eig
(
B
);
Dp
=
max
(
D
,
zeros
(
n
));
% positive eigenvalues
Dm
=
max
(
Dp
-
D
,
zeros
(
n
));
% negative eigenvalues
Bp
=
V
*
Dp
*
V
'
;
Bm
=
V
*
Dm
*
V
'
;
% if needed, matrix square root of Bp and Bm
%Rp = V*sqrtm(Dp);
%Rm = V*sqrtm(Dm);
Rp
=
V
*
sqrt
(
Dp
);
% Dp is diagonal !!!
Rm
=
V
*
sqrt
(
Dm
);
% Dm is diagonal !!!
%cvx_solver sdpt3;
cvx_solver
sedumi
;
cvx_begin
variable
X
(
n
,
n
)
variable
Yp
(
n
,
n
)
symmetric
variable
Ym
(
n
,
n
)
symmetric
% variable Zpl(n,n)
% variable Zm(n,n)
% variable Tp(2*n,2*n) symmetric
% variable Tm(2*n,2*n) symmetric
minimize
(
trace
(
A
*
(
Yp
-
Ym
))
)
subject
to
Yp
*
e
==
X
*
Bp
*
e
Ym
*
e
==
X
*
Bm
*
e
diag
(
Yp
)
==
X
*
diag
(
Bp
)
diag
(
Ym
)
==
X
*
diag
(
Bm
)
Yp
>=
min
(
min
(
Bp
))
*
E
Ym
>=
min
(
min
(
Bm
))
*
E
[
I
,(
X
*
Rp
)
'
;
X
*
Rp
,
Yp
]
==
semidefinite
(
2
*
n
)
[
I
,(
X
*
Rm
)
'
;
X
*
Rm
,
Ym
]
==
semidefinite
(
2
*
n
)
% Zpl == X*Rp
% Zm == X*Rm
% Tp == [I,Zpl';Zpl,Yp]
% Tm == [I,Zm';Zm,Ym]
% Tp == semidefinite(2*n)
% Tm == semidefinite(2*n)
Yp
==
semidefinite
(
n
)
Ym
==
semidefinite
(
n
)
X
>=
0
sum
(
X
)
==
1
sum
(
X
'
)
==
1
cvx_end
exercises/ex06/solutions/nug30.m
0 → 100644
View file @
dcbab056
B
=
[
0
1
2
3
4
5
1
2
3
4
5
6
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
;
1
0
1
2
3
4
2
1
2
3
4
5
3
2
3
4
5
6
4
3
4
5
6
7
5
4
5
6
7
8
;
2
1
0
1
2
3
3
2
1
2
3
4
4
3
2
3
4
5
5
4
3
4
5
6
6
5
4
5
6
7
;
3
2
1
0
1
2
4
3
2
1
2
3
5
4
3
2
3
4
6
5
4
3
4
5
7
6
5
4
5
6
;
4
3
2
1
0
1
5
4
3
2
1
2
6
5
4
3
2
3
7
6
5
4
3
4
8
7
6
5
4
5
;
5
4
3
2
1
0
6
5
4
3
2
1
7
6
5
4
3
2
8
7
6
5
4
3
9
8
7
6
5
4
;
1
2
3
4
5
6
0
1
2
3
4
5
1
2
3
4
5
6
2
3
4
5
6
7
3
4
5
6
7
8
;
2
1
2
3
4
5
1
0
1
2
3
4
2
1
2
3
4
5
3
2
3
4
5
6
4
3
4
5
6
7
;
3
2
1
2
3
4
2
1
0
1
2
3
3
2
1
2
3
4
4
3
2
3
4
5
5
4
3
4
5
6
;
4
3
2
1
2
3
3
2
1
0
1
2
4
3
2
1
2
3
5
4
3
2
3
4
6
5
4
3
4
5
;
5
4
3
2
1
2
4
3
2
1
0
1
5
4
3
2
1
2
6
5
4
3
2
3
7
6
5
4
3
4
;
6
5
4
3
2
1
5
4
3
2
1
0
6
5
4
3
2
1
7
6
5
4
3
2
8
7
6
5
4
3
;
2
3
4
5
6
7
1
2
3
4
5
6
0
1
2
3
4
5
1
2
3
4
5
6
2
3
4
5
6
7
;
3
2
3
4
5
6
2
1
2
3
4
5
1
0
1
2
3
4
2
1
2
3
4
5
3
2
3
4
5
6
;
4
3
2
3
4
5
3
2
1
2
3
4
2
1
0
1
2
3
3
2
1
2
3
4
4
3
2
3
4
5
;
5
4
3
2
3
4
4
3
2
1
2
3
3
2
1
0
1
2
4
3
2
1
2
3
5
4
3
2
3
4
;
6
5
4
3
2
3
5
4
3
2
1
2
4
3
2
1
0
1
5
4
3
2
1
2
6
5
4
3
2
3
;
7
6
5
4
3
2
6
5
4
3
2
1
5
4
3
2
1
0
6
5
4
3
2
1
7
6
5
4
3
2
;
3
4
5
6
7
8
2
3
4
5
6
7
1
2
3
4
5
6
0
1
2
3
4
5
1
2
3
4
5
6
;
4
3
4
5
6
7
3
2
3
4
5
6
2
1
2
3
4
5
1
0
1
2
3
4
2
1
2
3
4
5
;
5
4
3
4
5
6
4
3
2
3
4
5
3
2
1
2
3
4
2
1
0
1
2
3
3
2
1
2
3
4
;
6
5
4
3
4
5
5
4
3
2
3
4
4
3
2
1
2
3
3
2
1
0
1
2
4
3
2
1
2
3
;
7
6
5
4
3
4
6
5
4
3
2
3
5
4
3
2
1
2
4
3
2
1
0
1
5
4
3
2
1
2
;
8
7
6
5
4
3
7
6
5
4
3
2
6
5
4
3
2
1
5
4
3
2
1
0
6
5
4
3
2
1
;
4
5
6
7
8
9
3
4
5
6
7
8
2
3
4
5
6
7
1
2
3
4
5
6
0
1
2
3
4
5
;
5
4
5
6
7
8
4
3
4
5
6
7
3
2
3
4
5
6
2
1
2
3
4
5
1
0
1
2
3
4
;
6
5
4
5
6
7
5
4
3
4
5
6
4
3
2
3
4
5
3
2
1
2
3
4
2
1
0
1
2
3
;
7
6
5
4
5
6
6
5
4
3
4
5
5
4
3
2
3
4
4
3
2
1
2
3
3
2
1
0
1
2
;
8
7
6
5
4
5
7
6
5
4
3
4
6
5
4
3
2
3
5
4
3
2
1
2
4
3
2
1
0
1
;
9
8
7
6
5
4
8
7
6
5
4
3
7
6
5
4
3
2
6
5
4
3
2
1
5
4
3
2
1
0
];
A
=
[
0
3
2
0
0
2
10
5
0
5
2
5
0
0
2
0
5
6
3
0
...
1
10
0
10
2
1
1
1
0
1
;
3
0
4
0
10
4
0
0
2
2
1
0
5
0
0
0
0
2
0
1
...
6
1
0
1
2
2
5
1
10
5
;
2
4
0
3
4
0
5
5
5
1
4
1
0
4
0
4
0
6
3
2
...
5
5
2
1
0
0
3
1
0
2
;
0
0
3
0
0
0
0
2
2
0
6
0
2
5
2
5
1
1
1
1
...
2
2
4
0
2
0
2
2
5
5
;
0
10
4
0
0
5
2
0
0
0
0
2
0
0
0
0
2
1
0
0
...
2
0
5
1
0
2
1
0
2
1
;
2
4
0
0
5
0
1
2
2
1
4
10
10
2
5
5
0
5
0
0
...
0
10
0
0
0
4
0
10
1
1
;
10
0
5
0
2
1
0
10
10
5
10
10
6
0
0
10
2
1
10
1
...
5
5
2
3
5
0
2
0
1
3
;
5
0
5
2
0
2
10
0
1
3
5
0
0
0
2
4
5
2
10
6
...
0
5
5
2
5
0
5
5
0
2
;
0
2
5
2
0
2
10
1
0
10
2
1
5
2
0
3
0
2
0
0
...
4
0
5
2
0
5
2
2
5
2
;
5
2
1
0
0
1
5
3
10
0
5
5
6
0
1
5
5
0
5
2
...
3
5
0
5
2
10
10
1
5
2
;
2
1
4
6
0
4
10
5
2
5
0
0
0
1
2
1
0
2
0
0
...
0
6
6
0
4
5
3
2
2
10
;
5
0
1
0
2
10
10
0
1
5
0
0
5
5
2
0
0
0
0
2
...
0
4
5
10
1
0
0
0
0
1
;
0
5
0
2
0
10
6
0
5
6
0
5
0
2
0
4
2
2
1
0
...
6
2
1
5
5
0
0
1
5
5
;
0
0
4
5
0
2
0
0
2
0
1
5
2
0
2
1
0
5
3
10
...
0
0
4
2
0
0
4
2
5
5
;
2
0
0
2
0
5
0
2
0
1
2
2
0
2
0
4
5
1
0
1
...
0
5
0
2
0
0
5
1
1
0
;
0
0
4
5
0
5
10
4
3
5
1
0
4
1
4
0
0
3
0
2
...
2
0
2
0
5
0
5
2
5
10
;
5
0
0
1
2
0
2
5
0
5
0
0
2
0
5
0
0
2
2
0
...
0
0
6
5
3
5
0
0
5
1
;
6
2
6
1
1
5
1
2
2
0
2
0
2
5
1
3
2
0
5
1
...
2
10
10
4
0
0
5
0
0
0
;
3
0
3
1
0
0
10
10
0
5
0
0
1
3
0
0
2
5
0
0
...
5
5
1
0
5
2
1
2
10
10
;
0
1
2
1
0
0
1
6
0
2
0
2
0
10
1
2
0
1
0
0
...
5
2
1
3
1
5
6
5
5
3
;
1
6
5
2
2
0
5
0
4
3
0
0
6
0
0
2
0
2
5
5
...
0
4
0
1
0
0
0
5
0
0
;
10
1
5
2
0
10
5
5
0
5
6
4
2
0
5
0
0
10
5
2
...
4
0
5
0
4
4
5
0
2
5
;
0
0
2
4
5
0
2
5
5
0
6
5
1
4
0
2
6
10
1
1
...
0
5
0
0
4
4
1
0
2
2
;
10
1
1
0
1
0
3
2
2
5
0
10
5
2
2
0
5
4
0
3
...
1
0
0
0
5
5
0
1
0
0
;
2
2
0
2
0
0
5
5
0
2
4
1
5
0
0
5
3
0
5
1
...
0
4
4
5
0
1
0
10
1
0
;
1
2
0
0
2
4
0
0
5
10
5
0
0
0
0
0
5
0
2
5
...
0
4
4
5
1
0
0
0
0
0
;
1
5
3
2
1
0
2
5
2
10
3
0
0
4
5
5
0
5
1
6
...
0
5
1
0
0
0
0
0
0
10
;
1
1
1
2
0
10
0
5
2
1
2
0
1
2
1
2
0
0
2
5
...
5
0
0
1
10
0
0
0
2
2
;
0
10
0
5
2
1
1
0
5
5
2
0
5
5
1
5
5
0
10
5
...
0
2
2
0
1
0
0
2
0
2
;
1
5
2
5
1
1
3
2
2
2
10
1
5
5
0
10
1
0
10
3
...
0
5
2
0
0
0
10
2
2
0
];
% swap A and B
%C=A; A=B; B=C;
n
=
30
;
e
=
ones
(
n
,
1
);
E
=
e
*
e
'
;
I
=
eye
(
n
);
% split B = Bp - Bm, with Bp and Bm positive semidefinite
[
V
,
D
]
=
eig
(
B
);
Dp
=
max
(
D
,
zeros
(
n
));
% positive eigenvalues
Dm
=
max
(
Dp
-
D
,
zeros
(
n
));
% negative eigenvalues
Bp
=
V
*
Dp
*
V
'
;
Bm
=
V
*
Dm
*
V
'
;
% if needed, matrix square root of Bp and Bm
%Rp = V*sqrtm(Dp);
%Rm = V*sqrtm(Dm);
Rp
=
V
*
sqrt
(
Dp
);
% Dp is diagonal !!!
Rm
=
V
*
sqrt
(
Dm
);
% Dm is diagonal !!!
%cvx_solver sdpt3;
cvx_solver
sedumi
;
cvx_begin
variable
X
(
n
,
n
)
variable
Yp
(
n
,
n
)
symmetric
variable
Ym
(
n
,
n
)
symmetric
% variable Zpl(n,n)
% variable Zm(n,n)
% variable Tp(2*n,2*n) symmetric
% variable Tm(2*n,2*n) symmetric
minimize
(
trace
(
A
*
(
Yp
-
Ym
))
)
subject
to
Yp
*
e
==
X
*
Bp
*
e
Ym
*
e
==
X
*
Bm
*
e
diag
(
Yp
)
==
X
*
diag
(
Bp
)
diag
(
Ym
)
==
X
*
diag
(
Bm
)
Yp
>=
min
(
min
(
Bp
))
*
E
Ym
>=
min
(
min
(
Bm
))
*
E
[
I
,(
X
*
Rp
)
'
;
X
*
Rp
,
Yp
]
==
semidefinite
(
2
*
n
)
[
I
,(
X
*
Rm
)
'
;
X
*
Rm
,
Ym
]
==
semidefinite
(
2
*
n
)
% Zpl == X*Rp
% Zm == X*Rm
% Tp == [I,Zpl';Zpl,Yp]
% Tm == [I,Zm';Zm,Ym]
% Tp == semidefinite(2*n)
% Tm == semidefinite(2*n)
Yp
==
semidefinite
(
n
)
Ym
==
semidefinite
(
n
)
X
>=
0
sum
(
X
)
==
1
sum
(
X
'
)
==
1
cvx_end
exercises/ex06/solutions/sol06.txt
0 → 100644
View file @
dcbab056
|------------------|
| Solution of ex06 |
|------------------|
---
6.1
---
See nug12.m, nug30.m, tai30a.m and compare with the optimal solution or bounds form the website on QAPLIB.
---
6.2
---
LP: see cows.mod
Objective: 17200.
Robust formulation of Soyster: see cows_worst.mod.
Objective: 16381.
Robust formulation of Ben-Tal and Nemirovski: see cows_robust.mod.
Objective: 17138.6
exercises/ex06/solutions/tai30a.m
0 → 100644
View file @
dcbab056
B
=
[
0
27
56
28
64
98
69
69
93
35
41
67
12
33
1
71
72
26
86
67
67
24
10
46
56
56
69
53
48
74
;
27
0
43
94
29
64
97
38
12
99
51
87
76
71
47
47
9
71
20
81
99
45
56
15
99
64
57
84
83
61
;
56
43
0
25
41
8
46
52
22
33
68
52
28
75
99
6
85
65
47
90
24
79
70
28
19
87
80
4
21
34
;
28
94
25
0
12
20
67
24
94
10
64
52
92
49
68
65
66
28
81
17
58
99
90
70
17
57
37
2
37
3
;
64
29
41
12
0
43
20
68
35
60
23
48
40
10
42
83
82
27
49
15
34
50
42
26
33
23
16
69
97
13
;
98
64
8
20
43
0
35
80
40
55
30
22
76
55
56
91
74
82
96
2
13
4
4
35
48
29
42
56
3
30
;
69
97
46
67
20
35
0
60
81
37
42
3
17
25
37
26
88
95
55
53
62
22
44
86
43
43
40
36
53
34
;
69
38
52
24
68
80
60
0
59
43
50
58
62
43
9
22
64
46
68
53
8
30
30
92
6
13
95
76
81
91
;
93
12
22
94
35
40
81
59
0
37
78
90
64
49
46
19
60
93
35
47
69
54
87
12
39
33
54
12
10
4
;
35
99
33
10
60
55
37
43
37
0
88
54
46
82
84
8
29
10
92
62
62
74
48
22
85
23
3
30
12
98
;
41
51
68
64
23
30
42
50
78
88
0
69
29
61
34
53
98
94
33
77
31
54
71
78
8
78
50
76
56
80
;
67
87
52
52
48
22
3
58
90
54
69
0
72
26
20
57
39
68
55
71
19
32
87
41
94
21
21
20
61
13
;
12
76
28
92
40
76
17
62
64
46
29
72
0
5
46
97
61
8
92
33
73
0
16
73
74
44
55
96
67
94
;
33
71
75
49
10
55
25
43
49
82
61
26
5
0
83
28
22
78
55
89
11
99
84
56
30
90
87
80
20
66
;
1
47
99
68
42
56
37
9
46
84
34
20
46
83
0
59
93
79
80
28
68
99
54
69
99
1
49
63
23
33
;
71
47
6
65
83
91
26
22
19
8
53
57
97
28
59
0
99
40
29
60
95
28
44
30
88
66
9
41
3
4
;
72
9
85
66
82
74
88
64
60
29
98
39
61
22
93
99
0
63
61
87
34
28
55
63
10
78
17
90
0
66
;
26
71
65
28
27
82
95
46
93
10
94
68
8
78
79
40
63
0
62
30
76
0
91
62
73
38
49
85
86
88
;
86
20
47
81
49
96
55
68
35
92
33
55
92
55
80
29
61
62
0
13
71
46
75
98
53
52
10
84
70
44
;
67
81
90
17
15
2
53
53
47
62
77
71
33
89
28
60
87
30
13
0
8
52
59
48
85
29
94
79
4
85
;
67
99
24
58
34
13
62
8
69
62
31
19
73
11
68
95
34
76
71
8
0
31
54
95
75
81
11
56
38
95
;
24
45
79
99
50
4
22
30
54
74
54
32
0
99
99
28
28
0
46
52
31
0
37
67
54
88
93
53
44
68
;
10
56
70
90
42
4
44
30
87
48
71
87
16
84
54
44
55
91
75
59
54
37
0
58
98
55
84
76
19
46
;
46
15
28
70
26
35
86
92
12
22
78
41
73
56
69
30
63
62
98
48
95
67
58
0
89
89
5
23
63
19
;
56
99
19
17
33
48
43
6
39
85
8
94
74
30
99
88
10
73
53
85
75
54
98
89
0
53
20
47
17
66
;
56
64
87
57
23
29
43
13
33
23
78
21
44
90
1
66
78
38
52
29
81
88
55
89
53
0
60
86
14
52
;
69
57
80
37
16
42
40
95
54
3
50
21
55
87
49
9
17
49
10
94
11
93
84
5
20
60
0
27
77
5
;
53
84
4
2
69
56
36
76
12
30
76
20
96
80
63
41
90
85
84
79
56
53
76
23
47
86
27
0
37
27
;
48
83
21
37
97
3
53
81
10
12
56
61
67
20
23
3
0
86
70
4
38
44
19
63
17
14
77
37
0
53
;
74
61
34
3
13
30
34
91
4
98
80
13
94
66
33
4
66
88
44
85
95
68
46
19
66
52
5
27
53
0
];
A
=
[
0
21
95
82
56
41
6
25
10
4
63
6
44
40
75
79
0
89
35
9
1
85
84
12
0
26
91
11
35
82
;
21
0
26
69
56
86
45
91
59
18
76
39
18
57
36
61
36
21
71
11
29
82
82
6
71
8
77
74
30
89
;
95
26
0
76
76
40
93
56
1
50
4
36
27
85
2
1
15
11
35
11
20
21
61
80
58
21
76
72
44
85
;
82
69
76
0
94
90
51
3
48
29
90
66
41
15
83
96
74
45
65
40
54
83
14
71
77
36
53
37
26
87
;
56
56
76
94
0
76
91
13
29
11
77
32
87
67
94
79
2
10
99
56
70
99
60
4
56
2
60
72
74
46
;
41
86
40
90
76
0
13
20
86
4
77
15
89
48
14
89
44
59
22
57
63
6
0
62
41
62
46
25
75
76
;
6
45
93
51
91
13
0
40
66
58
30
68
78
91
13
59
49
85
84
8
38
41
56
39
53
77
50
30
58
55
;
25
91
56
3
13
20
40
0
19
85
52
34
53
40
69
12
85
72
7
49
46
87
58
17
68
27
21
6
67
26
;
10
59
1
48
29
86
66
19
0
82
44
35
3
62
8
51
1
91
39
87
72
45
96
7
87
68
33
3
21
90
;
4
18
50
29
11
4
58
85
82
0
45
47
25
30
43
97
33
35
61
42
36
43
7
84
6
0
0
48
62
59
;
63
76
4
90
77
77
30
52
44
45
0
29
94
82
29
3
3
51
67
39
15
66
42
23
62
62
28
76
66
82
;
6
39
36
66
32
15
68
34
35
47
29
0
98
35
15
17
77
44
26
76
86
60
62
62
83
91
57
62
36
2
;
44
18
27
41
87
89
78
53
3
25
94
98
0
2
43
65
37
49
61
5
34
53
96
82
48
28
31
75
1
95
;
40
57
85
15
67
48
91
40
62
30
82
35
2
0
7
92
69
62
32
97
5
39
50
82
93
71
35
14
20
74
;
75
36
2
83
94
14
13
69
8
43
29
15
43
7
0
49
50
37
79
19
51
70
42
26
79
98
60
35
9
96
;
79
61
1
96
79
89
59
12
51
97
3
17
65
92
49
0
70
21
37
37
67
93
93
39
2
52
26
90
26
1
;
0
36
15
74
2
44
49
85
1
33
3
77
37
69
50
70
0
68
93
7
94
19
54
37
0
20
12
11
66
84
;
89
21
11
45
10
59
85
72
91
35
51
44
49
62
37
21
68
0
80
1
55
9
21
12
65
7
17
51
84
87
;
35
71
35
65
99
22
84
7
39
61
67
26
61
32
79
37
93
80
0
2
27
82
71
71
40
93
27
93
92
34
;
9
11
11
40
56
57
8
49
87
42
39
76
5
97
19
37
7
1
2
0
39
31
26
1
87
72
59
97
46
62
;
1
29
20
54
70
63
38
46
72
36
15
86
34
5
51
67
94
55
27
39
0
12
91
63
70
1
22
49
24
58
;
85
82
21
83
99
6
41
87
45
43
66
60
53
39
70
93
19
9
82
31
12
0
62
49
94
92
63
13
45
22
;
84
82
61
14
60
0
56
58
96
7
42
62
96
50
42
93
54
21
71
26
91
62
0
69
70
18
1
44
32
3
;
12
6
80
71
4
62
39
17
7
84
23
62
82
82
26
39
37
12
71
1
63
49
69
0
72
99
34
45
18
96
;
0
71
58
77
56
41
53
68
87
6
62
83
48
93
79
2
0
65
40
87
70
94
70
72
0
82
79
75
83
43
;
26
8
21
36
2
62
77
27
68
0
62
91
28
71
98
52
20
7
93
72
1
92
18
99
82
0
26
81
39
66
;
91
77
76
53
60
46
50
21
33
0
28
57
31
35
60
26
12
17
27
59
22
63
1
34
79
26
0
22
71
58
;
11
74
72
37
72
25
30
6
3
48
76
62
75
14
35
90
11
51
93
97
49
13
44
45
75
81
22
0
42
91
;
35
30
44
26
74
75
58
67
21
62
66
36
1
20
9
26
66
84
92
46
24
45
32
18
83
39
71
42
0
56
;
82
89
85
87
46
76
55
26
90
59
82
2
95
74
96
1
84
87
34
62
58
22
3
96
43
66
58
91
56
0
];
% swap A and B
%C=A; A=B; B=C;
n
=
30
;
e
=
ones
(
n
,
1
);
E
=
e
*
e
'
;
I
=
eye
(
n
);
% split B = Bp - Bm, with Bp and Bm positive semidefinite
[
V
,
D
]
=
eig
(
B
);
Dp
=
max
(
D
,
zeros
(
n
));
% positive eigenvalues
Dm
=
max
(
Dp
-
D
,
zeros
(
n
));
% negative eigenvalues
Bp
=
V
*
Dp
*
V
'
;
Bm
=
V
*
Dm
*
V
'
;
% if needed, matrix square root of Bp and Bm
%Rp = V*sqrtm(Dp);
%Rm = V*sqrtm(Dm);
Rp
=
V
*
sqrt
(
Dp
);
% Dp is diagonal !!!
Rm
=
V
*
sqrt
(
Dm
);
% Dm is diagonal !!!
%cvx_solver sdpt3;
cvx_solver
sedumi
;
cvx_begin
variable
X
(
n
,
n
)
variable
Yp
(
n
,
n
)
symmetric
variable
Ym
(
n
,
n
)
symmetric
% variable Zpl(n,n)
% variable Zm(n,n)
% variable Tp(2*n,2*n) symmetric
% variable Tm(2*n,2*n) symmetric
minimize
(
trace
(
A
*
(
Yp
-
Ym
))
)
subject
to
Yp
*
e
==
X
*
Bp
*
e
Ym
*
e
==
X
*
Bm
*
e
diag
(
Yp
)
==
X
*
diag
(
Bp
)
diag
(
Ym
)
==
X
*
diag
(
Bm
)
Yp
>=
min
(
min
(
Bp
))
*
E
Ym
>=
min
(
min
(
Bm
))
*
E
[
I
,(
X
*
Rp
)
'
;
X
*
Rp
,
Yp
]
==
semidefinite
(
2
*
n
)
[
I
,(
X
*
Rm
)
'
;
X
*
Rm
,
Ym
]
==
semidefinite
(
2
*
n
)
% Zpl == X*Rp
% Zm == X*Rm
% Tp == [I,Zpl';Zpl,Yp]
% Tm == [I,Zm';Zm,Ym]
% Tp == semidefinite(2*n)
% Tm == semidefinite(2*n)
Yp
==
semidefinite
(
n
)
Ym
==
semidefinite
(
n
)
X
>=
0
sum
(
X
)
==
1
sum
(
X
'
)
==
1
cvx_end
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