randn('state',0);
m = 16; n = 8;
A = randn(m,n);
b = randn(m,1);
B = randn(m,n);
disp('Computing optimal solution for 1st formulation...');
cvx_begin
variable x1(n)
minimize( matrix_frac(A*x1 + b , eye(m) + B*diag(x1)*B') )
x1 >= 0;
cvx_end
opt1 = cvx_optval;
disp('Computing optimal solution for 2nd formulation...');
cvx_begin
variable x2(n)
variable Y(n,n) diagonal
minimize( matrix_frac(A*x2 + b , eye(m) + B*Y*B') )
x2 >= 0;
Y == diag(x2);
cvx_end
opt2 = cvx_optval;
disp('Computing optimal solution for 3rd formulation...');
cvx_begin
variables x3(n) w(n) v(m)
minimize( square_pos(norm(v)) + matrix_frac(w, diag(x3)) )
v + B*w == A*x3 + b;
x3 >= 0;
cvx_end
opt3 = cvx_optval;
disp('Computing optimal solution for 4th formulation...');
cvx_begin
variables x4(n) w(n) v(m)
variable Y(n,n) diagonal
minimize( square_pos(norm(v)) + matrix_frac(w, Y) )
v + B*w == A*x4 + b;
x4 >= 0;
Y == diag(x4);
cvx_end
opt4 = cvx_optval;
disp('------------------------------------------------------------------------');
disp('The optimal value for each of the 4 formulations is: ');
[opt1 opt2 opt3 opt4]
disp('They should be equal!')
Computing optimal solution for 1st formulation...
Calling sedumi: 161 variables, 9 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 9, order n = 26, dim = 298, blocks = 2
nnz(A) = 1225 + 0, nnz(ADA) = 81, nnz(L) = 45
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 3.39E+01 0.000
1 : -2.62E+00 9.95E+00 0.000 0.2932 0.9000 0.9000 1.31 1 1 1.2E+01
2 : -3.95E+00 3.75E+00 0.000 0.3767 0.9000 0.9000 1.25 1 1 4.0E+00
3 : -4.65E+00 1.03E+00 0.000 0.2736 0.9000 0.9000 1.25 1 1 9.4E-01
4 : -4.97E+00 3.95E-01 0.000 0.3851 0.9000 0.9000 0.95 1 1 3.8E-01
5 : -5.14E+00 8.42E-02 0.000 0.2131 0.9000 0.9000 1.01 1 1 8.1E-02
6 : -5.18E+00 6.78E-03 0.093 0.0806 0.9900 0.9900 1.02 1 1 6.4E-03
7 : -5.18E+00 2.58E-04 0.363 0.0381 0.9901 0.9900 1.00 1 1 2.8E-04
8 : -5.18E+00 1.09E-05 0.251 0.0421 0.9000 0.0000 1.00 1 1 5.8E-05
9 : -5.18E+00 8.97E-07 0.000 0.0824 0.9121 0.9000 1.00 1 1 6.7E-06
10 : -5.18E+00 1.19E-07 0.000 0.1321 0.9097 0.9000 1.00 1 1 9.9E-07
11 : -5.18E+00 1.97E-08 0.000 0.1665 0.9111 0.9000 1.00 1 1 1.8E-07
12 : -5.18E+00 1.16E-09 0.000 0.0588 0.9901 0.9900 1.00 1 1 1.1E-08
iter seconds digits c*x b*y
12 0.1 Inf -5.1824773412e+00 -5.1824772175e+00
|Ax-b| = 5.4e-09, [Ay-c]_+ = 1.9E-08, |x|= 6.9e+00, |y|= 5.2e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 6.000E-02 1.000E-02
Max-norms: ||b||=1, ||c|| = 2.976981e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 3.52372.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.18248
Computing optimal solution for 2nd formulation...
Calling sedumi: 161 variables, 9 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 9, order n = 26, dim = 298, blocks = 2
nnz(A) = 1225 + 0, nnz(ADA) = 81, nnz(L) = 45
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 3.39E+01 0.000
1 : -2.62E+00 9.95E+00 0.000 0.2932 0.9000 0.9000 1.31 1 1 1.2E+01
2 : -3.95E+00 3.75E+00 0.000 0.3767 0.9000 0.9000 1.25 1 1 4.0E+00
3 : -4.65E+00 1.03E+00 0.000 0.2736 0.9000 0.9000 1.25 1 1 9.4E-01
4 : -4.97E+00 3.95E-01 0.000 0.3851 0.9000 0.9000 0.95 1 1 3.8E-01
5 : -5.14E+00 8.42E-02 0.000 0.2131 0.9000 0.9000 1.01 1 1 8.1E-02
6 : -5.18E+00 6.78E-03 0.093 0.0806 0.9900 0.9900 1.02 1 1 6.4E-03
7 : -5.18E+00 2.58E-04 0.363 0.0381 0.9901 0.9900 1.00 1 1 2.8E-04
8 : -5.18E+00 1.09E-05 0.251 0.0421 0.9000 0.0000 1.00 1 1 5.8E-05
9 : -5.18E+00 8.97E-07 0.000 0.0824 0.9121 0.9000 1.00 1 1 6.7E-06
10 : -5.18E+00 1.19E-07 0.000 0.1321 0.9097 0.9000 1.00 1 1 9.9E-07
11 : -5.18E+00 1.97E-08 0.000 0.1665 0.9111 0.9000 1.00 1 1 1.8E-07
12 : -5.18E+00 1.16E-09 0.000 0.0588 0.9901 0.9900 1.00 1 1 1.1E-08
iter seconds digits c*x b*y
12 0.1 Inf -5.1824773412e+00 -5.1824772175e+00
|Ax-b| = 5.4e-09, [Ay-c]_+ = 1.9E-08, |x|= 6.9e+00, |y|= 5.2e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 7.000E-02 1.000E-02
Max-norms: ||b||=1, ||c|| = 2.976981e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 3.52372.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.18248
Computing optimal solution for 3rd formulation...
Calling sedumi: 75 variables, 54 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 54, order n = 24, dim = 113, blocks = 4
nnz(A) = 321 + 0, nnz(ADA) = 2740, nnz(L) = 1397
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 4.11E+00 0.000
1 : 1.90E+00 1.88E+00 0.000 0.4561 0.9000 0.9000 0.91 1 1 6.7E+00
2 : 2.55E+00 4.95E-01 0.000 0.2638 0.9000 0.9000 1.40 1 1 1.5E+00
3 : 4.09E+00 1.29E-01 0.000 0.2612 0.9000 0.9000 1.02 1 1 3.8E-01
4 : 4.86E+00 3.41E-02 0.000 0.2635 0.9000 0.9000 0.97 1 1 1.0E-01
5 : 5.12E+00 6.90E-03 0.000 0.2024 0.9000 0.9000 1.00 1 1 2.0E-02
6 : 5.17E+00 5.29E-04 0.000 0.0766 0.8425 0.9000 0.99 1 1 4.4E-03
7 : 5.18E+00 9.99E-05 0.000 0.1891 0.9000 0.9000 1.00 1 1 8.4E-04
8 : 5.18E+00 1.89E-05 0.000 0.1891 0.9000 0.9000 1.00 1 1 1.6E-04
9 : 5.18E+00 3.75E-06 0.000 0.1982 0.9000 0.9000 1.00 1 1 3.2E-05
10 : 5.18E+00 7.68E-07 0.102 0.2050 0.9000 0.9000 1.00 1 1 6.5E-06
11 : 5.18E+00 1.61E-07 0.177 0.2097 0.9000 0.9000 1.00 1 1 1.4E-06
12 : 5.18E+00 3.41E-08 0.218 0.2117 0.9000 0.9000 1.00 1 1 2.9E-07
13 : 5.18E+00 7.29E-09 0.223 0.2138 0.9000 0.9000 1.00 1 1 6.1E-08
14 : 5.18E+00 1.54E-09 0.226 0.2112 0.9000 0.9000 1.00 2 2 1.3E-08
iter seconds digits c*x b*y
14 0.1 Inf 5.1824773736e+00 5.1824773780e+00
|Ax-b| = 2.4e-08, [Ay-c]_+ = 1.3E-09, |x|= 6.2e+00, |y|= 1.1e+01
Detailed timing (sec)
Pre IPM Post
0.000E+00 1.000E-01 1.000E-02
Max-norms: ||b||=1.488490e+00, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 282957.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.18248
Computing optimal solution for 4th formulation...
Calling sedumi: 75 variables, 54 equality constraints
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 54, order n = 24, dim = 113, blocks = 4
nnz(A) = 321 + 0, nnz(ADA) = 2740, nnz(L) = 1397
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 4.11E+00 0.000
1 : 1.90E+00 1.88E+00 0.000 0.4561 0.9000 0.9000 0.91 1 1 6.7E+00
2 : 2.55E+00 4.95E-01 0.000 0.2638 0.9000 0.9000 1.40 1 1 1.5E+00
3 : 4.09E+00 1.29E-01 0.000 0.2612 0.9000 0.9000 1.02 1 1 3.8E-01
4 : 4.86E+00 3.41E-02 0.000 0.2635 0.9000 0.9000 0.97 1 1 1.0E-01
5 : 5.12E+00 6.90E-03 0.000 0.2024 0.9000 0.9000 1.00 1 1 2.0E-02
6 : 5.17E+00 5.29E-04 0.000 0.0766 0.8425 0.9000 0.99 1 1 4.4E-03
7 : 5.18E+00 9.99E-05 0.000 0.1891 0.9000 0.9000 1.00 1 1 8.4E-04
8 : 5.18E+00 1.89E-05 0.000 0.1891 0.9000 0.9000 1.00 1 1 1.6E-04
9 : 5.18E+00 3.75E-06 0.000 0.1982 0.9000 0.9000 1.00 1 1 3.2E-05
10 : 5.18E+00 7.68E-07 0.102 0.2050 0.9000 0.9000 1.00 1 1 6.5E-06
11 : 5.18E+00 1.61E-07 0.177 0.2097 0.9000 0.9000 1.00 1 1 1.4E-06
12 : 5.18E+00 3.41E-08 0.218 0.2115 0.9000 0.9000 1.00 1 1 2.9E-07
13 : 5.18E+00 7.27E-09 0.223 0.2134 0.9000 0.9000 1.00 1 1 6.1E-08
14 : 5.18E+00 1.49E-09 0.225 0.2054 0.9000 0.9000 1.00 1 2 1.3E-08
iter seconds digits c*x b*y
14 0.1 Inf 5.1824773737e+00 5.1824773781e+00
|Ax-b| = 2.4e-08, [Ay-c]_+ = 1.3E-09, |x|= 6.2e+00, |y|= 1.2e+01
Detailed timing (sec)
Pre IPM Post
1.000E-02 1.000E-01 0.000E+00
Max-norms: ||b||=1.488490e+00, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 318280.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.18248
------------------------------------------------------------------------
The optimal value for each of the 4 formulations is:
ans =
5.1825 5.1825 5.1825 5.1825
They should be equal!