randn('state',0);
n = 10;
W = randn(n); W = 0.5*(W + W');
fprintf(1,'Solving the dual of the two-way partitioning problem...');
cvx_begin sdp
variable nu(n)
maximize ( -sum(nu) )
W + diag(nu) >= 0;
cvx_end
fprintf(1,'Done! \n');
opt1 = cvx_optval;
fprintf(1,'Solving the SDP relaxation of the two-way partitioning problem...');
cvx_begin sdp
variable X(n,n) symmetric
minimize ( trace(W*X) )
diag(X) == 1;
X >= 0;
cvx_end
fprintf(1,'Done! \n');
opt2 = cvx_optval;
disp('------------------------------------------------------------------------');
disp('The optimal value of the Lagrange dual and the SDP relaxation fo the ');
disp('two-way partitioning problem are, respectively, ');
disp([opt1 opt2])
disp('They are equal as expected!');
Solving the dual of the two-way partitioning problem...
Calling sedumi: 55 variables, 10 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 = 10, order n = 11, dim = 101, blocks = 2
nnz(A) = 10 + 0, nnz(ADA) = 100, nnz(L) = 55
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.36E+00 0.000
1 : -1.28E+01 4.80E-01 0.000 0.3538 0.9000 0.9000 0.88 1 1 1.1E+00
2 : -2.51E+01 1.08E-01 0.000 0.2254 0.9000 0.9000 0.68 1 1 2.8E-01
3 : -2.80E+01 2.00E-02 0.000 0.1851 0.9000 0.9000 0.95 1 1 5.4E-02
4 : -2.88E+01 1.25E-03 0.000 0.0624 0.9900 0.9900 0.98 1 1 3.4E-03
5 : -2.88E+01 2.71E-04 0.000 0.2167 0.9000 0.9000 1.00 1 1 7.3E-04
6 : -2.88E+01 1.25E-06 0.212 0.0046 0.9900 0.9784 1.00 1 1 4.0E-05
7 : -2.88E+01 1.14E-07 0.000 0.0914 0.9900 0.9900 1.00 1 1 3.7E-06
8 : -2.88E+01 1.02E-08 0.457 0.0887 0.9900 0.9900 1.00 1 1 3.3E-07
9 : -2.88E+01 9.45E-10 0.311 0.0930 0.9900 0.9900 1.00 1 1 3.1E-08
10 : -2.88E+01 2.84E-10 0.059 0.3002 0.9000 0.9000 1.00 2 2 9.2E-09
iter seconds digits c*x b*y
10 0.1 Inf -2.8825674846e+01 -2.8825674817e+01
|Ax-b| = 8.6e-09, [Ay-c]_+ = 2.1E-08, |x|= 8.6e+00, |y|= 1.0e+01
Detailed timing (sec)
Pre IPM Post
0.000E+00 5.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 2.873183e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 604.228.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -26.6924
Done!
Solving the SDP relaxation of the two-way partitioning problem...
Calling sedumi: 55 variables, 10 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 = 10, order n = 11, dim = 101, blocks = 2
nnz(A) = 10 + 0, nnz(ADA) = 100, nnz(L) = 55
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.36E+00 0.000
1 : -1.28E+01 4.80E-01 0.000 0.3538 0.9000 0.9000 0.88 1 1 1.1E+00
2 : -2.51E+01 1.08E-01 0.000 0.2254 0.9000 0.9000 0.68 1 1 2.8E-01
3 : -2.80E+01 2.00E-02 0.000 0.1851 0.9000 0.9000 0.95 1 1 5.4E-02
4 : -2.88E+01 1.25E-03 0.000 0.0624 0.9900 0.9900 0.98 1 1 3.4E-03
5 : -2.88E+01 2.71E-04 0.000 0.2167 0.9000 0.9000 1.00 1 1 7.3E-04
6 : -2.88E+01 1.25E-06 0.212 0.0046 0.9900 0.9784 1.00 1 1 4.0E-05
7 : -2.88E+01 1.14E-07 0.000 0.0914 0.9900 0.9900 1.00 1 1 3.7E-06
8 : -2.88E+01 1.02E-08 0.457 0.0887 0.9900 0.9900 1.00 1 1 3.3E-07
9 : -2.88E+01 9.45E-10 0.311 0.0930 0.9900 0.9900 1.00 1 1 3.1E-08
10 : -2.88E+01 2.84E-10 0.059 0.3002 0.9000 0.9000 1.00 2 2 9.2E-09
iter seconds digits c*x b*y
10 0.0 Inf -2.8825674846e+01 -2.8825674817e+01
|Ax-b| = 8.6e-09, [Ay-c]_+ = 2.1E-08, |x|= 8.6e+00, |y|= 1.0e+01
Detailed timing (sec)
Pre IPM Post
1.000E-02 5.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 2.873183e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 604.228.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -26.6924
Done!
------------------------------------------------------------------------
The optimal value of the Lagrange dual and the SDP relaxation fo the
two-way partitioning problem are, respectively,
-26.6924 -26.6924
They are equal as expected!