Reference

Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology,Springer Optimization and Its Applications, Model Surface (3.10) in chapter Some Mathematical Algorithms and Problems in GAMS Technology, 2013

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Minimal surface problem.
Find a function f that minimizes the array of its graph subject to some
constraints on the boundary of the domain of f.

Boyd, S., Vandenberghe, L., Convex Optimization, Cambridge University Press,
Cambridge, 2004.
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SET X /I1*I21/;
SET Y /J1*J21/;
SET inside(X,Y);

* Exclude i1 and i21 from inside
inside(X,Y)$(not((ord(X)=1) and (ord(X)=card(X)))) = yes;

display inside;

SCALAR K /10/;

VARIABLES     obj, f(x,y);
POSITIVE VARIABLE  f(X,Y);

* Bounds on variables, initial conditions, fixing conditions:
f.up(x,y)=1;
f.l(x,y) =1.0;
f.fx(X,Y)$((ord(X)=1) or (ord(X)=card(X))) = 1;

EQUATION objfun;

objfun.. obj =E= (1/sqr(K)) *
                  sum((X,Y) $(inside(X,Y)),
                       sqrt( sqr((F(X+1,Y)-F(X,Y))/K) +
                             sqr((F(X,Y+1)-F(X,Y))/K) + 1) ) ;

MODEL surface /all/;

SOLVE surface using nlp minimizing obj;

$ifThenI x%mode%==xbook
file res1 /surf1.dat/
put res1;
put "Array surface =" obj.l; put /;
loop(Y, put Y.tl:6; loop(X, put F.l(X,Y):6:2 ); put /;) put /;
$endIf
* End surface