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 Mosek 9.1.9: 351 variables, 150 equality constraints
------------------------------------------------------------
MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 150
Cones : 81
Scalar variables : 351
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 150
Cones : 81
Scalar variables : 351
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 96
Optimizer - Cones : 81
Optimizer - Scalar variables : 297 conic : 243
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 520 after factor : 564
Factor - dense dim. : 0 flops : 7.15e+03
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.0e+00 1.0e+00 5.5e+01 0.00e+00 2.700000000e+01 -2.700000000e+01 1.0e+00 0.00
1 5.0e-01 2.5e-01 1.0e+01 7.27e-02 1.257717072e+01 -6.681174583e+00 2.5e-01 0.01
2 1.1e-01 5.6e-02 1.7e+00 4.18e-01 1.208648815e+01 5.852200415e+00 5.6e-02 0.01
3 4.1e-02 2.0e-02 4.5e-01 4.48e-01 1.642111580e+01 1.351707810e+01 2.0e-02 0.01
4 1.6e-02 8.2e-03 1.3e-01 6.64e-01 1.887884124e+01 1.754780456e+01 8.2e-03 0.01
5 3.4e-03 1.7e-03 1.2e-02 8.82e-01 2.028591708e+01 1.999663914e+01 1.7e-03 0.01
6 1.1e-03 5.5e-04 2.3e-03 9.77e-01 2.053657001e+01 2.044223025e+01 5.5e-04 0.01
7 3.1e-04 1.6e-04 3.5e-04 9.93e-01 2.061430046e+01 2.058749786e+01 1.6e-04 0.01
8 1.0e-04 5.2e-05 6.7e-05 9.99e-01 2.063618935e+01 2.062728005e+01 5.2e-05 0.01
9 3.2e-06 1.6e-06 3.7e-07 9.99e-01 2.064600292e+01 2.064572783e+01 1.6e-06 0.01
10 2.6e-08 1.3e-08 2.6e-10 1.00e+00 2.064632086e+01 2.064631868e+01 1.3e-08 0.01
11 2.4e-09 1.2e-09 7.4e-12 1.00e+00 2.064632335e+01 2.064632315e+01 1.2e-09 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 2.0646323351e+01 nrm: 2e+00 Viol. con: 8e-09 var: 4e-09 cones: 0e+00
Dual. obj: 2.0646323148e+01 nrm: 7e+00 Viol. con: 0e+00 var: 3e-09 cones: 0e+00
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 11 time: 0.01
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +20.6463
Done!