Reference

Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology,Springer Optimization and Its Applications, Model Minsurf (3.12) in chapter Some Mathematical Algorithms and Problems in GAMS Technology, 2013

$onText
Find the surface with minimal area that lies above an obstacle with given
boundary conditions.

This model is from the COPS benchmarking suite.

See http://www-unix.mcs.anl.gov/~more/cops/.

The number of internal grid points can be specified using the command line
parameters --nx and --ny.

COPS performance tests have been reported for nx-1 = 50, ny-1 = 25, 50, 75, 100

Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS.
Tech. rep., Mathematics and Computer Science Division, 2000.

Friedman, A, Free Boundary Problems in Science and Technology.
Notices Amer. Math. Soc. 47 (2000), 854-861.
$offText

$if not set nx $set nx 70
$if not set ny $set ny 70

sets nx grid points in 1st direction / x0*x%nx% /
     ny grid points in 2st direction / y0*y%ny% /

alias(nx,i),(ny,j);

parameters hx grid spacing for x
           hy grid spacing for y
           area area of triangle;

hx := 1/(card(nx)-1);
hy := 1/(card(ny)-1);
area := 0.5*hx*hy;

variables v(nx,ny) defines the finite element approximation
          surf;
positive variable v;

equation defsurf;

defsurf..
surf/area =e= sum((nx(i+1),ny(j+1)),
         sqrt(1+sqr((v[i+1,j]-v[i,j])/hx)+sqr((v[i,j+1]-v[i,j])/hy))) +
              sum((nx(i-1),ny(j-1)),
         sqrt(1+sqr((v[i-1,j]-v[i,j])/hx)+sqr((v[i,j-1]-v[i,j])/hy)));

v.fx['x0' ,j] = 0;
v.fx['x%nx%',j] = 0;
v.fx[i,'y0' ] = 1 - sqr(2*(ord(i)-1)*hx-1);
v.fx[i,'y%ny%'] = 1 - sqr(2*(ord(i)-1)*hx-1);

v.lo(i,j)$(((ord(i)-1)  >= floor(0.25/hx) and
            (ord(i)-1)  <= ceil(0.75/hx)) and
            ((ord(j)-1) >= floor(0.25/hy) and
            (ord(j)-1)  <= ceil(0.75/hy))) = 1;
v.l(i,j) = 1 - sqr(2*(ord(i)-1)*hx-1);

model minsurf / all /;

$ifThenI x%mode%==xbook
$onEcho >minos.opt
  superbasic limit 5000
$offEcho
minsurf.workspace=125;
minsurf.reslim=6000;
$endIf

solve minsurf minimizing surf using nlp;

$ifThenI x%mode%==xbook
file rez /minsurf.dat/
put rez;
loop(i, loop(j, put v.l(i,j):6:2); put/;);put/;
$endIf

* End Minsurf