```% Section 8.7.3, Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 10/24/05
%
% K fixed points x_1,...,x_K in R^2 are given and the goal is to place
% one additional point x such that the sum of the squares of the
% Euclidean distances to fixed points is minimized:
%           minimize    sum_{i=1}^K  ||x - x_i||^2
% The optimal point is the average of the given fixed points

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

% minimizing the sum of Euclidean distance
fprintf(1,'Minimizing the sum of the squares the distances to fixed points...');

cvx_begin
variable x(2)
minimize ( sum( square_pos( norms(x*ones(1,K) - P,2) ) ) )
cvx_end

fprintf(1,'Done! \n');

% Displaying results
disp('------------------------------------------------------------------');
disp('The optimal point location is: ');
disp(x);
disp('The average location of the fixed points is');
disp(sum(P,2)/K);
disp('They are the same as expected!');
```
```Minimizing the sum of the squares the distances to fixed points...
Calling SDPT3 4.0: 88 variables, 42 equality constraints
------------------------------------------------------------

num. of constraints = 42
dim. of sdp    var  = 22,   num. of sdp  blk  = 11
dim. of socp   var  = 33,   num. of socp blk  = 11
dim. of linear var  = 22
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version  predcorr  gam  expon  scale_data
HKM      1      0.000   1        0
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
0|0.000|0.000|1.8e+01|1.5e+01|7.3e+03| 1.100000e+02  0.000000e+00| 0:0:00| chol  1  1
1|0.637|0.756|6.7e+00|3.8e+00|3.5e+03| 1.418911e+02 -6.667195e+01| 0:0:00| chol  1  1
2|0.763|0.990|1.6e+00|4.6e-02|9.9e+02| 2.987271e+02 -1.488760e+02| 0:0:00| chol  1  1
3|1.000|1.000|5.9e-07|1.0e-03|1.6e+02| 1.005763e+02 -6.360729e+01| 0:0:00| chol  1  1
4|1.000|1.000|9.4e-08|1.0e-04|4.9e+01| 4.828646e+01 -8.200827e-01| 0:0:00| chol  1  1
5|0.885|1.000|1.2e-08|1.0e-05|8.6e+00| 2.267532e+01  1.408310e+01| 0:0:00| chol  1  1
6|1.000|0.967|1.5e-09|1.3e-06|2.3e+00| 1.824747e+01  1.593419e+01| 0:0:00| chol  1  1
7|0.906|0.979|4.2e-10|1.3e-07|2.6e-01| 1.688821e+01  1.662966e+01| 0:0:00| chol  1  1
8|1.000|0.982|6.2e-10|1.2e-08|5.5e-02| 1.672161e+01  1.666619e+01| 0:0:00| chol  1  1
9|0.970|0.978|1.4e-10|1.4e-09|1.8e-03| 1.668444e+01  1.668266e+01| 0:0:00| chol  1  1
10|0.976|0.983|3.2e-12|5.1e-11|3.9e-05| 1.668315e+01  1.668311e+01| 0:0:00| chol  1  1
11|0.926|1.000|1.5e-11|1.0e-12|7.0e-06| 1.668312e+01  1.668312e+01| 0:0:00| chol  2  1
12|1.000|1.000|4.9e-11|1.5e-12|8.0e-07| 1.668312e+01  1.668312e+01| 0:0:00| chol  1  2
13|0.994|1.000|1.5e-12|2.3e-12|1.6e-08| 1.668312e+01  1.668312e+01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations   = 13
primal objective value =  1.66831189e+01
dual   objective value =  1.66831189e+01
gap := trace(XZ)       = 1.55e-08
relative gap           = 4.52e-10
actual relative gap    = 4.51e-10
rel. primal infeas (scaled problem)   = 1.54e-12
rel. dual     "        "       "      = 2.25e-12
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual     "        "       "      = 0.00e+00
norm(X), norm(y), norm(Z) = 1.1e+01, 1.3e+01, 1.9e+01
norm(A), norm(b), norm(C) = 1.0e+01, 9.0e+00, 4.3e+00
Total CPU time (secs)  = 0.25
CPU time per iteration = 0.02
termination code       =  0
DIMACS: 2.9e-12  0.0e+00  4.9e-12  0.0e+00  4.5e-10  4.5e-10
-------------------------------------------------------------------

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +16.6831

Done!
------------------------------------------------------------------
The optimal point location is:
0.0379
0.0785

The average location of the fixed points is
0.0379
0.0785

They are the same as expected!
```