linewidth = 1;
markersize = 5;
fixed = [ 1 1 -1 -1 1 -1 -0.2 0.1;
1 -1 -1 1 -0.5 -0.2 -1 1]';
M = size(fixed,1);
N = 6;
A = [ 1 0 0 -1 0 0 0 0 0 0 0 0 0 0
1 0 -1 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 -1 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 -1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 -1 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 -1 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 -1
0 1 -1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 -1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 -1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 -1 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 -1 0
0 0 1 -1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 -1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 -1 0 0 0
0 0 0 1 -1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 -1 0 0 0 0 0
0 0 0 1 0 0 0 0 0 -1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 -1 0 0
0 0 0 1 0 -1 0 0 0 0 0 -1 0 0
0 0 0 0 1 -1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 -1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 -1 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 -1
0 0 0 0 0 1 0 0 -1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 -1 0 0 0 ];
nolinks = size(A,1);
fprintf(1,'Computing the optimal locations of the 6 free points...');
cvx_begin
variable x(N+M,2)
minimize ( sum(norms( A*x,2,2 )))
x(N+[1:M],:) == fixed;
cvx_end
fprintf(1,'Done! \n');
free_sum = x(1:N,:);
figure(1);
dots = plot(free_sum(:,1), free_sum(:,2), 'or', fixed(:,1), fixed(:,2), 'bs');
set(dots(1),'MarkerFaceColor','red');
hold on
legend('Free points','Fixed points','Location','Best');
for i=1:nolinks
ind = find(A(i,:));
line2 = plot(x(ind,1), x(ind,2), ':k');
hold on
set(line2,'LineWidth',linewidth);
end
axis([-1.1 1.1 -1.1 1.1]) ;
axis equal;
title('Linear placement problem');
figure(2)
all = [free_sum; fixed];
bins = 0.05:0.1:1.95;
lengths = sqrt(sum((A*all).^2')');
[N2,hist2] = hist(lengths,bins);
bar(hist2,N2);
hold on;
xx = linspace(0,2,1000); yy = 2*xx;
plot(xx,yy,'--');
axis([0 2 0 4.5]);
hold on
plot([0 2], [0 0 ], 'k-');
title('Distribution of the 27 link lengths');
Computing the optimal locations of the 6 free points...
Calling sedumi: 81 variables, 39 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 = 39, order n = 55, dim = 82, blocks = 28
nnz(A) = 101 + 0, nnz(ADA) = 279, nnz(L) = 167
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 3.44E-01 0.000
1 : -1.40E+01 8.21E-02 0.000 0.2387 0.9000 0.9000 3.01 1 1 6.5E-01
2 : -2.01E+01 1.82E-02 0.000 0.2217 0.9000 0.9000 1.13 1 1 1.4E-01
3 : -2.16E+01 3.56E-03 0.000 0.1955 0.9000 0.9000 1.02 1 1 2.7E-02
4 : -2.18E+01 7.01E-04 0.000 0.1969 0.9000 0.9000 1.00 1 1 5.2E-03
5 : -2.19E+01 6.81E-05 0.165 0.0971 0.9900 0.9900 1.00 1 1 5.1E-04
6 : -2.19E+01 1.20E-06 0.198 0.0177 0.9900 0.9236 1.00 1 1 4.6E-05
7 : -2.19E+01 6.13E-08 0.136 0.0509 0.9900 0.9900 1.00 1 1 2.4E-06
8 : -2.19E+01 1.48E-08 0.000 0.2415 0.9009 0.9000 1.00 1 1 5.7E-07
9 : -2.19E+01 1.32E-09 0.401 0.0893 0.9900 0.9699 1.00 1 1 5.1E-08
10 : -2.19E+01 4.18E-10 0.000 0.3166 0.9275 0.9000 1.00 2 2 1.5E-08
iter seconds digits c*x b*y
10 0.1 Inf -2.1908263691e+01 -2.1908263520e+01
|Ax-b| = 1.1e-08, [Ay-c]_+ = 1.3E-08, |x|= 7.3e+00, |y|= 5.2e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 6.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 2,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +21.9083
Done!