n = 50; threshold = 0.2529;
rand('state',209);
xy = rand(n,2);
angle = 10*pi/180;
Rotate = [ cos(angle) sin(angle); -sin(angle) cos(angle) ];
xy = (Rotate*xy')';
Dist = zeros(n,n);
for i=1:(n-1);
for j=i+1:n;
Dist(i,j) = norm( xy(i,:) - xy(j,:) );
end;
end;
Dist = Dist + Dist';
Ad = Dist < threshold;
Ad = Ad - eye(n);
m = sum(sum(Ad))/2;
A = zeros(n,m);
l = 0;
for i=1:(n-1);
for j=i+1:n;
if Ad(i,j)>0.5
l = l + 1;
A(i,l) = 1;
A(j,l) = -1;
end;
end;
end;
A = sparse(A);
[n,m] = size(A);
[ w_fdla, rho_fdla ] = fdla(A);
[ w_fmmc, rho_fmmc ] = fmmc(A);
[ w_md, rho_md ] = max_deg(A);
[ w_bc, rho_bc ] = best_const(A);
[ w_mh, rho_mh ] = mh(A);
tau_fdla = 1/log(1/rho_fdla);
tau_fmmc = 1/log(1/rho_fmmc);
tau_md = 1/log(1/rho_md);
tau_bc = 1/log(1/rho_bc);
tau_mh = 1/log(1/rho_mh);
eig_opt = sort(eig(eye(n) - A * diag(w_fdla) * A'));
eig_fmmc = sort(eig(eye(n) - A * diag(w_fmmc) * A'));
eig_mh = sort(eig(eye(n) - A * diag(w_mh) * A'));
eig_md = sort(eig(eye(n) - A * diag(w_md) * A'));
eig_bc = sort(eig(eye(n) - A * diag(w_bc) * A'));
fprintf(1,'\nResults:\n');
fprintf(1,'FDLA weights:\t\t rho = %5.4f \t tau = %5.4f\n',rho_fdla,tau_fdla);
fprintf(1,'FMMC weights:\t\t rho = %5.4f \t tau = %5.4f\n',rho_fmmc,tau_fmmc);
fprintf(1,'M-H weights:\t\t rho = %5.4f \t tau = %5.4f\n',rho_mh,tau_mh);
fprintf(1,'MAX_DEG weights:\t rho = %5.4f \t tau = %5.4f\n',rho_md,tau_md);
fprintf(1,'BEST_CONST weights:\t rho = %5.4f \t tau = %5.4f\n',rho_bc,tau_bc);
figure(1), clf
gplot(Ad,xy);
hold on;
plot(xy(:,1), xy(:,2), 'ko','LineWidth',4, 'MarkerSize',4);
axis([0.05 1.1 -0.1 0.95]);
title('Graph')
hold off;
figure(2), clf
v_fdla = [w_fdla; diag(eye(n) - A*diag(w_fdla)*A')];
[ifdla, jfdla, neg_fdla] = find( v_fdla.*(v_fdla < -0.001 ) );
v_fdla(ifdla) = [];
wbins = [-0.6:0.012:0.6];
hist(neg_fdla,wbins); hold on,
h = findobj(gca,'Type','patch');
set(h,'FaceColor','r')
hist(v_fdla,wbins); hold off,
axis([-0.6 0.6 0 12]);
xlabel('optimal FDLA weights');
ylabel('histogram');
figure(3), clf
xbins = (-1:0.015:1)';
ymax = 6;
subplot(3,1,1)
hist(eig_md, xbins); hold on;
max_md = max(abs(eig_md(1:n-1)));
plot([-max_md -max_md],[0 ymax], 'b--');
plot([ max_md max_md],[0 ymax], 'b--');
axis([-1 1 0 ymax]);
text(0,5,'MAX DEG');
title('Eigenvalue distributions')
subplot(3,1,2)
hist(eig_bc, xbins); hold on;
max_opt = max(abs(eig_bc(1:n-1)));
plot([-max_opt -max_opt],[0 ymax], 'b--');
plot([ max_opt max_opt],[0 ymax], 'b--');
axis([-1 1 0 ymax]);
text(0,5,'BEST CONST');
subplot(3,1,3)
hist(eig_opt, xbins); hold on;
max_opt = max(abs(eig_opt(1:n-1)));
plot([-max_opt -max_opt],[0 ymax], 'b--');
plot([ max_opt max_opt],[0 ymax], 'b--');
axis([-1 1 0 ymax]);
text(0,5,'FDLA');
figure(4), clf
xbins = (-1:0.015:1)';
ymax = 6;
subplot(3,1,1)
hist(eig_md, xbins); hold on;
max_md = max(abs(eig_md(1:n-1)));
plot([-max_md -max_md],[0 ymax], 'b--');
plot([ max_md max_md],[0 ymax], 'b--');
axis([-1 1 0 ymax]);
text(0,5,'MAX DEG');
title('Eigenvalue distributions')
subplot(3,1,2)
hist(eig_mh, xbins); hold on;
max_opt = max(abs(eig_mh(1:n-1)));
plot([-max_opt -max_opt],[0 ymax], 'b--');
plot([ max_opt max_opt],[0 ymax], 'b--');
axis([-1 1 0 ymax]);
text(0,5,'MH');
subplot(3,1,3)
hist(eig_fmmc, xbins); hold on;
max_opt = max(abs(eig_fmmc(1:n-1)));
plot([-max_opt -max_opt],[0 ymax], 'b--');
plot([ max_opt max_opt],[0 ymax], 'b--');
axis([-1 1 0 ymax]);
text(0,5,'FMMC');
figure(5), clf
v_fmmc = [w_fmmc; diag(eye(n) - A*diag(w_fmmc)*A')];
[ifmmc, jfmmc, nonzero_fmmc] = find( v_fmmc.*(v_fmmc > 0.001 ) );
hist(nonzero_fmmc,80);
axis([0 1 0 10]);
xlabel('optimal positive FMMC weights');
ylabel('histogram');
figure(6), clf
An = abs(A*diag(w_fmmc)*A');
An = (An - diag(diag(An))) > 0.0001;
gplot(An,xy,'b-'); hold on;
h = findobj(gca,'Type','line');
set(h,'LineWidth',2.5)
gplot(Ad,xy,'b:');
plot(xy(:,1), xy(:,2), 'ko','LineWidth',4, 'MarkerSize',4);
axis([0.05 1.1 -0.1 0.95]);
title('Subgraph with positive transition prob.')
hold off;
Calling sedumi: 2551 variables, 2350 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
Split 1 free variables
eqs m = 2350, order n = 103, dim = 5003, blocks = 3
nnz(A) = 3203 + 0, nnz(ADA) = 3318750, nnz(L) = 1660550
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.96E-01 0.000
1 : 3.60E+00 2.38E-02 0.000 0.0804 0.9900 0.9900 -0.22 1 1 8.8E-01
2 : 1.12E+00 9.42E-03 0.000 0.3954 0.9000 0.9000 3.06 1 1 1.5E-01
3 : 9.43E-01 3.44E-03 0.000 0.3655 0.9000 0.9000 1.86 1 1 4.1E-02
4 : 9.27E-01 1.01E-03 0.000 0.2938 0.9000 0.9000 1.09 1 1 1.2E-02
5 : 9.08E-01 3.29E-04 0.000 0.3250 0.9000 0.9000 1.06 1 1 3.9E-03
6 : 9.04E-01 1.05E-04 0.000 0.3193 0.9000 0.9000 1.02 1 1 1.3E-03
7 : 9.03E-01 2.58E-05 0.000 0.2460 0.9022 0.9000 1.00 1 1 3.4E-04
8 : 9.02E-01 4.63E-06 0.000 0.1796 0.9074 0.9000 1.00 1 1 9.1E-05
9 : 9.02E-01 5.57E-07 0.000 0.1201 0.9187 0.9000 1.00 1 1 2.3E-05
10 : 9.02E-01 1.12E-07 0.000 0.2011 0.9160 0.9000 1.00 1 1 5.4E-06
11 : 9.02E-01 2.56E-08 0.000 0.2284 0.9055 0.9000 1.00 1 1 1.3E-06
12 : 9.02E-01 5.43E-09 0.000 0.2123 0.9054 0.9000 1.00 1 1 2.7E-07
13 : 9.02E-01 1.49E-09 0.000 0.2756 0.9063 0.9000 1.00 1 1 7.6E-08
14 : 9.02E-01 3.41E-10 0.000 0.2279 0.9027 0.9000 1.00 1 1 1.7E-08
15 : 9.02E-01 6.99E-11 0.000 0.2051 0.9000 0.9031 1.00 1 2 3.6E-09
iter seconds digits c*x b*y
15 20.0 Inf 9.0207867670e-01 9.0207869224e-01
|Ax-b| = 3.2e-09, [Ay-c]_+ = 1.5E-09, |x|= 1.1e+01, |y|= 1.2e+00
Detailed timing (sec)
Pre IPM Post
3.300E-01 2.003E+01 1.000E-02
Max-norms: ||b||=9.400000e-01, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 54.6293.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.902079
Calling sedumi: 2801 variables, 2600 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
Split 1 free variables
eqs m = 2600, order n = 353, dim = 5253, blocks = 3
nnz(A) = 3702 + 0, nnz(ADA) = 3918898, nnz(L) = 1961925
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.59E-02 0.000
1 : 1.13E+00 5.63E-02 0.000 0.6559 0.9000 0.9000 5.60 1 1 1.7E+00
2 : 1.06E+00 2.85E-02 0.000 0.5056 0.9000 0.9000 1.54 1 1 8.3E-01
3 : 9.31E-01 8.13E-03 0.000 0.2855 0.9000 0.9000 1.80 1 1 1.6E-01
4 : 9.53E-01 1.71E-03 0.000 0.2103 0.9000 0.9000 1.48 1 1 2.7E-02
5 : 9.39E-01 4.28E-04 0.000 0.2503 0.9000 0.9000 1.12 1 1 6.5E-03
6 : 9.29E-01 2.20E-04 0.000 0.5135 0.9000 0.9000 1.06 1 1 3.3E-03
7 : 9.24E-01 1.24E-04 0.000 0.5648 0.9000 0.9000 1.04 1 1 1.9E-03
8 : 9.24E-01 3.10E-05 0.383 0.2500 0.9000 0.0000 1.03 1 1 1.4E-03
9 : 9.22E-01 7.28E-06 0.012 0.2347 0.9480 0.9000 1.02 1 1 4.9E-04
10 : 9.18E-01 2.65E-06 0.000 0.3643 0.9000 0.9071 1.02 1 1 1.8E-04
11 : 9.17E-01 1.39E-06 0.000 0.5231 0.9255 0.9000 1.01 1 1 9.1E-05
12 : 9.16E-01 8.29E-07 0.000 0.5969 0.9468 0.9000 1.01 1 1 5.3E-05
13 : 9.16E-01 4.24E-07 0.225 0.5120 0.0000 0.9000 1.00 1 1 2.9E-05
14 : 9.15E-01 1.45E-07 0.000 0.3429 0.9000 0.9098 1.00 1 1 1.0E-05
15 : 9.15E-01 5.76E-08 0.000 0.3957 0.8138 0.9000 1.00 1 1 4.2E-06
16 : 9.15E-01 1.96E-08 0.000 0.3407 0.9000 0.8329 1.00 1 1 1.4E-06
17 : 9.15E-01 6.23E-09 0.000 0.3176 0.8328 0.9000 1.00 1 1 4.5E-07
18 : 9.15E-01 1.51E-09 0.000 0.2427 0.9000 0.9000 1.00 1 1 1.1E-07
19 : 9.15E-01 1.17E-10 0.000 0.0774 0.9900 0.9900 1.00 1 2 8.5E-09
iter seconds digits c*x b*y
19 35.0 Inf 9.1515175405e-01 9.1515175583e-01
|Ax-b| = 6.9e-09, [Ay-c]_+ = 4.9E-09, |x|= 1.1e+01, |y|= 1.2e+00
Detailed timing (sec)
Pre IPM Post
4.200E-01 3.498E+01 2.000E-02
Max-norms: ||b||=9.600000e-01, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 81.924.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.915152
Results:
FDLA weights: rho = 0.9021 tau = 9.7037
FMMC weights: rho = 0.9152 tau = 11.2784
M-H weights: rho = 0.9489 tau = 19.0839
MAX_DEG weights: rho = 0.9706 tau = 33.5236
BEST_CONST weights: rho = 0.9470 tau = 18.3549