Figure 8.17: Fourth-order placement problem

% Section 8.7.3, Boyd & Vandenberghe "Convex Optimization"
% Original by Lieven Vandenberghe
% Adapted for CVX by Joelle Skaf - 10/24/05
% (a figure is generated)
%
% Placement problem with 6 free points, 8 fixed points and 27 links.
% The coordinates of the free points minimize the sum of the squares of
% Euclidean lengths of the links, i.e.
%           minimize    sum_{i<j) h(||x_i - x_j||)
% where h(z) = z^4.

linewidth = 1;      % in points;  width of dotted lines
markersize = 5;    % in points;  marker size

fixed = [ 1   1  -1 -1    1   -1  -0.2  0.1; % coordinates of fixed points
          1  -1  -1  1 -0.5 -0.2    -1    1]';
M = size(fixed,1);  % number of fixed points
N = 6;              % number of free points

% first N columns of A correspond to free points,
% last M columns correspond to fixed points

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        % error in data!!!
      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);    % number of links

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');

% Plots
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');
% print -deps placement-quartic.eps

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');
% print -deps placement-quartic-hist.eps
Computing the optimal locations of the 6 free points... 
Calling sedumi: 297 variables, 147 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 = 147, order n = 217, dim = 352, blocks = 82
nnz(A) = 317 + 0, nnz(ADA) = 805, nnz(L) = 484
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            1.10E+01 0.000
  1 :  -2.70E+00 3.15E+00 0.000 0.2863 0.9000 0.9000   3.05  1  1  1.5E+00
  2 :  -7.56E+00 9.54E-01 0.000 0.3029 0.9000 0.9000   0.92  1  1  7.2E-01
  3 :  -1.35E+01 3.33E-01 0.000 0.3492 0.9000 0.9000   0.53  1  1  2.7E-01
  4 :  -1.78E+01 1.08E-01 0.000 0.3252 0.9000 0.9000   0.65  1  1  9.8E-02
  5 :  -2.00E+01 2.27E-02 0.000 0.2095 0.9000 0.9000   0.84  1  1  2.2E-02
  6 :  -2.05E+01 6.39E-03 0.000 0.2816 0.9000 0.9000   0.97  1  1  6.3E-03
  7 :  -2.05E+01 7.97E-06 0.101 0.0012 0.9000 0.0000   0.99  1  1  2.2E-03
  8 :  -2.06E+01 1.54E-06 0.000 0.1931 0.9168 0.9000   1.00  1  1  4.7E-04
  9 :  -2.06E+01 8.40E-08 0.000 0.0546 0.9187 0.9900   1.00  1  1  2.9E-05
 10 :  -2.06E+01 3.64E-09 0.000 0.0433 0.9900 0.9615   1.00  1  1  1.3E-06
 11 :  -2.06E+01 2.75E-10 0.366 0.0755 0.9479 0.9900   1.00  1  1  9.5E-08
 12 :  -2.06E+01 1.00E-11 0.000 0.0364 0.9900 0.9900   1.00  1  1  3.5E-09

iter seconds digits       c*x               b*y
 12      0.1   Inf -2.0646323157e+01 -2.0646323148e+01
|Ax-b| =   3.5e-09, [Ay-c]_+ =   3.9E-09, |x|=  4.1e+01, |y|=  8.5e+00

Detailed timing (sec)
   Pre          IPM          Post
1.000E-02    9.000E-02    0.000E+00    
Max-norms: ||b||=1, ||c|| = 2,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2.97702.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +20.6463
Done!