Example 8.4: One free point localization
n = 2;
K = 11;
randn('state',0);
P = randn(n,K);
fprintf(1,'Minimizing the L1-norm of the sum of the distances to fixed points...');
cvx_begin
variable x1(2)
minimize ( sum(norms(x1*ones(1,K) - P,1)) )
cvx_end
fprintf(1,'Done! \n');
fprintf(1,'Minimizing the L2-norm of the sum of the distances to fixed points...');
cvx_begin
variable x2(2)
minimize ( sum(norms(x2*ones(1,K) - P,2)) )
cvx_end
fprintf(1,'Done! \n');
disp('------------------------------------------------------------------');
disp('The optimal point location for the L1-norm case is: ');
disp(x1);
disp('The optimal point location for the L2-norm case is: ');
disp(x2);
Minimizing the L1-norm of the sum of the distances to fixed points...
Calling sedumi: 44 variables, 20 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 = 20, order n = 45, dim = 45, blocks = 23
nnz(A) = 40 + 0, nnz(ADA) = 200, nnz(L) = 110
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 6.16E+01 0.000
1 : 9.09E+00 1.93E+01 0.000 0.3131 0.9000 0.9000 3.59 1 1 8.8E-01
2 : 1.22E+01 4.58E+00 0.000 0.2376 0.9000 0.9000 1.24 1 1 2.3E-01
3 : 1.35E+01 9.57E-01 0.000 0.2088 0.9000 0.9000 1.04 1 1 5.2E-02
4 : 1.37E+01 2.75E-01 0.000 0.2878 0.9000 0.9000 1.01 1 1 1.5E-02
5 : 1.39E+01 1.65E-02 0.000 0.0598 0.9900 0.9900 1.00 1 1 9.7E-04
6 : 1.39E+01 7.99E-07 0.215 0.0000 1.0000 1.0000 1.00 1 1 4.7E-08
7 : 1.39E+01 2.67E-10 0.026 0.0003 0.9999 0.9999 1.00 1 1 1.7E-11
iter seconds digits c*x b*y
7 0.0 10.7 1.3868099975e+01 1.3868099974e+01
|Ax-b| = 1.0e-11, [Ay-c]_+ = 2.5E-12, |x|= 5.8e+00, |y|= 4.2e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 3.000E-02 0.000E+00
Max-norms: ||b||=3.848770e+00, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +13.8681
Done!
Minimizing the L2-norm of the sum of the distances to fixed points...
Calling sedumi: 33 variables, 13 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 = 13, order n = 23, dim = 34, blocks = 12
nnz(A) = 33 + 0, nnz(ADA) = 59, nnz(L) = 36
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 9.24E-01 0.000
1 : -4.97E+00 2.59E-01 0.000 0.2808 0.9000 0.9000 2.94 1 1 7.2E-01
2 : -9.93E+00 5.34E-02 0.000 0.2060 0.9000 0.9000 1.12 1 1 1.4E-01
3 : -1.14E+01 4.34E-03 0.000 0.0812 0.9900 0.9900 1.01 1 1 1.2E-02
4 : -1.15E+01 2.74E-04 0.000 0.0631 0.9900 0.9900 1.00 1 1 7.3E-04
5 : -1.15E+01 2.60E-05 0.318 0.0950 0.9900 0.9900 1.00 1 1 6.9E-05
6 : -1.15E+01 1.08E-06 0.000 0.0416 0.9000 0.5609 1.00 1 1 2.6E-05
7 : -1.15E+01 5.54E-08 0.455 0.0512 0.9900 0.9761 1.00 1 1 1.3E-06
8 : -1.15E+01 5.28E-09 0.204 0.0952 0.9900 0.9900 1.00 1 1 1.3E-07
9 : -1.15E+01 1.48E-09 0.000 0.2811 0.9000 0.9000 1.00 2 2 3.6E-08
10 : -1.15E+01 3.00E-10 0.000 0.2022 0.9047 0.9000 1.00 2 2 7.2E-09
iter seconds digits c*x b*y
10 0.1 Inf -1.1483929221e+01 -1.1483929196e+01
|Ax-b| = 7.0e-09, [Ay-c]_+ = 9.6E-09, |x|= 4.7e+00, |y|= 4.5e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 6.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 3.848770e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +11.4839
Done!
------------------------------------------------------------------
The optimal point location for the L1-norm case is:
-0.0956
0.1139
The optimal point location for the L2-norm case is:
0.1251
0.1716