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: 46 variables (2 free), 22 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 2 free variables
eqs m = 22, order n = 49, dim = 49, blocks = 1
nnz(A) = 88 + 0, nnz(ADA) = 242, nnz(L) = 132
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 7.19E+001 0.000
1 : 7.88E+000 1.87E+001 0.000 0.2598 0.9000 0.9000 2.59 1 1 1.0E+000
2 : 1.19E+001 4.29E+000 0.000 0.2297 0.9000 0.9000 1.15 1 1 2.9E-001
3 : 1.34E+001 8.96E-001 0.000 0.2087 0.9000 0.9000 1.02 1 1 6.7E-002
4 : 1.37E+001 2.54E-001 0.000 0.2835 0.9000 0.9000 1.00 1 1 1.9E-002
5 : 1.39E+001 1.55E-002 0.000 0.0611 0.9900 0.9900 1.00 1 1 1.2E-003
6 : 1.39E+001 8.05E-007 0.209 0.0001 1.0000 1.0000 1.00 1 1
iter seconds digits c*x b*y
6 0.0 Inf 1.3868099974e+001 1.3868099974e+001
|Ax-b| = 1.4e-015, [Ay-c]_+ = 2.7E-016, |x|= 5.6e+000, |y|= 4.5e+000
Detailed timing (sec)
Pre IPM Post
0.000E+000 0.000E+000 0.000E+000
Max-norms: ||b||=2.183186e+000, ||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: 35 variables (2 free), 22 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 2 free variables
eqs m = 22, order n = 27, dim = 38, blocks = 12
nnz(A) = 66 + 0, nnz(ADA) = 264, nnz(L) = 208
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.04E+001 0.000
1 : 9.28E+000 6.74E+000 0.000 0.3301 0.9000 0.9000 2.25 1 1 1.1E+000
2 : 1.08E+001 1.38E+000 0.000 0.2040 0.9000 0.9000 1.30 1 1 2.3E-001
3 : 1.14E+001 1.08E-001 0.000 0.0788 0.9900 0.9900 1.04 1 1 1.8E-002
4 : 1.15E+001 7.38E-003 0.000 0.0681 0.9900 0.9900 1.00 1 1 1.3E-003
5 : 1.15E+001 3.90E-004 0.000 0.0529 0.9900 0.9900 1.00 1 1 6.7E-005
6 : 1.15E+001 6.24E-005 0.000 0.1598 0.9073 0.9000 1.00 1 1 1.1E-005
7 : 1.15E+001 7.13E-006 0.212 0.1144 0.9455 0.9450 1.00 1 1 1.3E-006
8 : 1.15E+001 1.58E-006 0.000 0.2221 0.9036 0.9000 1.00 1 1 2.9E-007
9 : 1.15E+001 2.61E-007 0.000 0.1645 0.9058 0.9000 1.00 2 2 5.1E-008
10 : 1.15E+001 4.14E-008 0.000 0.1587 0.9055 0.9000 1.00 2 2 9.1E-009
iter seconds digits c*x b*y
10 0.1 8.3 1.1483929305e+001 1.1483929241e+001
|Ax-b| = 2.6e-009, [Ay-c]_+ = 1.3E-009, |x|= 6.9e+000, |y|= 3.3e+000
Detailed timing (sec)
Pre IPM Post
0.000E+000 6.009E-002 0.000E+000
Max-norms: ||b||=2.183186e+000, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 83.9825.
------------------------------------------------------------------------
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