* Cute AMPL model  (translation to GAMS)
*
* Objective  function: convex nonlinear
* Constraint functions: none

* Computes the median of m numbers by minimizing
* the sum of the distances.  When m is odd, median
* lies at the middle data point.  The corresponding
* distance term becomes almost "singular" but doesn't
* create any problems.

$Set M 19
Set i /i1*i%M%/;

Parameter a(i) ; a(i) = Uniform(0,1) ;

Variable epsi , x ,  sum_dists;  ;

Equation   Def_obj ;
Def_obj..  sum_dists =e= epsi + sum{i, sqrt(sqr(epsi)+sqr(x-a[i])) };

epsi.lo = 1.0e-8;
epsi.l = 1 ;

x.l = 0.5;

Model socp_vareps /all/;

Solve socp_vareps  using nlp minimize sum_dists;

display   a           ;
display   x.l         ;
display   sum_dists.l ;