Figure 8.16: Quadratic placement problem
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(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('Quadratic 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 = (4/1.5^2)*xx.^2;
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: 189 variables, 93 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 = 93, order n = 136, dim = 217, blocks = 55
nnz(A) = 209 + 0, nnz(ADA) = 589, nnz(L) = 349
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 5.28E+00 0.000
1 : -5.94E+00 1.52E+00 0.000 0.2873 0.9000 0.9000 2.61 1 1 1.8E+00
2 : -1.44E+01 4.31E-01 0.000 0.2838 0.9000 0.9000 0.81 1 1 4.4E-01
3 : -1.91E+01 8.89E-02 0.000 0.2063 0.9000 0.9000 0.85 1 1 9.4E-02
4 : -2.05E+01 3.43E-03 0.000 0.0386 0.9900 0.9900 0.96 1 1 3.7E-03
5 : -2.05E+01 7.55E-05 0.000 0.0220 0.9900 0.9900 1.00 1 1 8.2E-05
6 : -2.05E+01 1.51E-08 0.370 0.0002 0.9835 0.9900 1.00 1 1 1.8E-06
7 : -2.05E+01 9.64E-10 0.000 0.0637 0.9900 0.9900 1.00 1 1 1.1E-07
8 : -2.05E+01 7.75E-11 0.393 0.0804 0.9900 0.9900 1.00 1 1 9.0E-09
iter seconds digits c*x b*y
8 0.1 Inf -2.0547313255e+01 -2.0547313238e+01
|Ax-b| = 2.0e-08, [Ay-c]_+ = 6.6E-09, |x|= 1.8e+01, |y|= 6.7e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 5.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 2,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2.67068.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +20.5473
Done!