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(square_pos(square_pos(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('Fourth-order 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 = (6/1.5^4)*xx.^4;
plot(xx,yy,'--');
axis([0 1.5 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: 351 variables, 150 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 = 150, order n = 271, dim = 406, blocks = 82
nnz(A) = 374 + 0, nnz(ADA) = 1160, nnz(L) = 691
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 5.75E+00 0.000
1 : 4.85E+00 1.63E+00 0.000 0.2830 0.9000 0.9000 3.05 1 1 1.8E+00
2 : 8.35E+00 4.76E-01 0.000 0.2927 0.9000 0.9000 0.94 1 1 6.6E-01
3 : 1.35E+01 1.71E-01 0.000 0.3585 0.9000 0.9000 0.51 1 1 3.0E-01
4 : 1.75E+01 5.92E-02 0.000 0.3472 0.9000 0.9000 0.62 1 1 1.3E-01
5 : 1.97E+01 1.69E-02 0.000 0.2851 0.9000 0.9000 0.81 1 1 4.0E-02
6 : 2.04E+01 4.87E-03 0.000 0.2884 0.9000 0.9000 0.94 1 1 1.2E-02
7 : 2.06E+01 1.60E-03 0.000 0.3282 0.9000 0.9000 0.99 1 1 3.9E-03
8 : 2.06E+01 4.03E-06 0.000 0.0025 0.9000 0.9069 1.00 1 1 7.2E-04
9 : 2.06E+01 7.11E-07 0.000 0.1761 0.9000 0.9053 1.00 1 1 1.3E-04
10 : 2.06E+01 2.55E-08 0.000 0.0358 0.9900 0.9902 1.00 1 1 4.5E-06
11 : 2.06E+01 9.10E-10 0.492 0.0357 0.9900 0.9903 1.00 1 1 1.5E-07
12 : 2.06E+01 2.10E-10 0.257 0.2306 0.9000 0.0000 1.00 2 2 4.1E-08
13 : 2.06E+01 4.93E-11 0.000 0.2349 0.8113 0.9000 1.00 2 2 9.6E-09
iter seconds digits c*x b*y
13 0.2 Inf 2.0646323095e+01 2.0646323101e+01
|Ax-b| = 9.3e-08, [Ay-c]_+ = 0.0E+00, |x|= 1.5e+01, |y|= 2.8e+01
Detailed timing (sec)
Pre IPM Post
0.000E+00 2.000E-01 0.000E+00
Max-norms: ||b||=2, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 11.1466.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +20.6463
Done!