Example 8.4: One free point localization

% Section 8.7.1, Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 10/23/05
%
% K fixed points (u1,v1),..., (uK,vK) in R^2 are given and the goal is to place
% one additional point (u,v) such that:
% 1) the L1-norm is minimized, i.e.
%           minimize    sum_{i=1}^K ( |u - u_i| + |v - v_i| )
%    the solution in this case is any median of the fixed points
% 2) the L2-norm is minimized, i.e.
%           minimize    sum_{i=1}^K ( |u - u_i|^2 + |v - v_i|^2 )^.5
%    the solution in this case is the Weber point of the fixed points

% Data generation
n = 2;
K = 11;
randn('state',0);
P = randn(n,K);

% L1 - norm
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');

% L2 - norm
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');

% Displaying results
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 SDPT3: 46 variables, 22 equality constraints
------------------------------------------------------------

 num. of constraints = 22
 dim. of socp   var  = 44,   num. of socp blk  = 22
 dim. of free   var  =  2 *** convert ublk to lblk
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      mean(obj)   cputime
-------------------------------------------------------------------
 0|0.000|0.000|8.0e-01|4.2e+00|3.8e+02| 2.475938e+01| 0:0:00| chol  1  1 
 1|1.000|0.814|6.8e-07|8.6e-01|9.2e+01| 3.250087e+01| 0:0:00| chol  1  1 
 2|1.000|0.994|2.2e-07|1.4e-02|9.9e+00| 1.555407e+01| 0:0:00| chol  1  1 
 3|0.853|0.559|3.7e-07|6.8e-03|2.6e+00| 1.351595e+01| 0:0:00| chol  1  1 
 4|1.000|0.241|8.4e-07|5.2e-03|1.5e+00| 1.334302e+01| 0:0:00| chol  1  1 
 5|0.831|0.588|1.5e-07|2.2e-03|6.0e-01| 1.364061e+01| 0:0:00| chol  1  1 
 6|0.999|0.788|9.0e-10|4.6e-04|1.2e-01| 1.381465e+01| 0:0:00| chol  1  1 
 7|0.985|0.982|1.8e-10|8.4e-06|2.2e-03| 1.386713e+01| 0:0:00| chol  1  1 
 8|0.989|0.989|3.8e-11|7.9e-06|2.9e-05| 1.386809e+01| 0:0:00| chol  1  1 
 9|1.000|0.989|1.0e-12|1.1e-07|4.7e-07| 1.386810e+01| 0:0:00| chol  1  1 
10|0.549|0.945|4.7e-13|1.7e-09|7.0e-08| 1.386810e+01| 0:0:00|
  stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
 number of iterations   = 10
 primal objective value =  1.38681000e+01
 dual   objective value =  1.38680999e+01
 gap := trace(XZ)       = 7.00e-08
 relative gap           = 2.44e-09
 actual relative gap    = 2.37e-09
 rel. primal infeas     = 4.71e-13
 rel. dual   infeas     = 1.72e-09
 norm(X), norm(y), norm(Z) = 5.8e+00, 4.5e+00, 6.5e+00
 norm(A), norm(b), norm(C) = 9.1e+00, 5.1e+00, 5.7e+00
 Total CPU time (secs)  = 0.2  
 CPU time per iteration = 0.0  
 termination code       =  0
 DIMACS: 7.5e-13  0.0e+00  4.9e-09  0.0e+00  2.4e-09  2.4e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +13.8681
Done! 
Minimizing the L2-norm of the sum of the distances to fixed points... 
Calling SDPT3: 33 variables, 13 equality constraints
   For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

 num. of constraints = 13
 dim. of socp   var  = 33,   num. of socp blk  = 11
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      mean(obj)   cputime
-------------------------------------------------------------------
 0|0.000|0.000|5.6e-01|3.4e+00|9.7e+01| 0.000000e+00| 0:0:00| chol  1  1 
 1|1.000|0.929|1.1e-07|3.0e-01|1.0e+01|-9.009752e+00| 0:0:00| chol  1  1 
 2|0.926|0.858|2.5e-07|4.8e-02|1.4e+00|-1.103897e+01| 0:0:00| chol  1  1 
 3|0.862|0.969|8.5e-08|2.1e-03|1.7e-01|-1.143853e+01| 0:0:00| chol  1  1 
 4|0.954|0.964|9.6e-08|1.4e-04|7.3e-03|-1.148135e+01| 0:0:00| chol  1  1 
 5|0.948|0.980|2.3e-08|9.1e-06|3.1e-04|-1.148377e+01| 0:0:00| chol  1  1 
 6|0.923|0.965|6.4e-09|3.3e-07|2.2e-05|-1.148392e+01| 0:0:00| chol  1  1 
 7|0.871|0.978|1.6e-09|8.6e-09|3.1e-06|-1.148393e+01| 0:0:00| chol  1  2 
 8|0.822|0.989|4.5e-10|4.1e-10|4.8e-07|-1.148393e+01| 0:0:00| chol  2  2 
 9|0.955|0.985|4.1e-11|9.7e-11|3.0e-08|-1.148393e+01| 0:0:00|
  stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
 number of iterations   =  9
 primal objective value = -1.14839292e+01
 dual   objective value = -1.14839293e+01
 gap := trace(XZ)       = 3.00e-08
 relative gap           = 1.25e-09
 actual relative gap    = 1.18e-09
 rel. primal infeas     = 4.06e-11
 rel. dual   infeas     = 9.70e-11
 norm(X), norm(y), norm(Z) = 4.7e+00, 4.1e+00, 5.8e+00
 norm(A), norm(b), norm(C) = 6.7e+00, 4.3e+00, 5.1e+00
 Total CPU time (secs)  = 0.2  
 CPU time per iteration = 0.0  
 termination code       =  0
 DIMACS: 8.8e-11  0.0e+00  1.6e-10  0.0e+00  1.2e-09  1.3e-09
-------------------------------------------------------------------
------------------------------------------------------------
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: 
    0.1252
    0.1716