Tuesday, August 27, 2013

2D Interpolation With SOS2 variables

Using SOS2 variables to implement a 1D interpolation scheme is fairly easy (see: http://yetanothermathprogrammingconsultant.blogspot.com/2009/06/gams-piecewise-linear-functions-with.html). However, a 2D problem is already much more difficult. Here is an example taken from the Lindo web site:

$ontext

 
2D Interpolation with SOS2 variables

 
See: http://www.lindo.com/cgi-bin/modelf.cgi?Piecelin2Dsos3.txt;LINGO

$offtext


$set  n1 3
$set  n2 3
$eval n3 (%n1%+%n2%-1)


sets
   i
/i1*i%n1%/
   j
/j1*j%n2%/
   k
/k1*k%n3%/
;

table data(i,j,*)
          
x   y   f

i1.j1    195 1800  20
i1.j2    217 1900  26
i1.j3    240 2000  30
i2.j1    195 3500  52
i2.j2    217 3600  61
i2.j3    240 4100  78
i3.j1    195 5100  69
i3.j2    217 5200  80
i3.j3    240 5600  93
;

parameters
   xv(i,j)
   yv(i,j)
   fv(i,j)
;
xv(i,j) = data(i,j,
'x');
yv(i,j) = data(i,j,
'y'
);
fv(i,j) = data(i,j,
'f'
);

sos2 variables

  wx(i)
  wy(j)
  wd(k)
;



positive variables
  WGT(i,j)
  xa
  ya
  fa
;


variables z;

equations

   xconvex
   yconvex
   dconvex
   ewx
   ewy
   ewd

   compx
   compy
   compfv

   obj
;

xconvex..
sum(i, wx(i)) =e= 1;
yconvex..
sum
(j, wy(j)) =e= 1;
dconvex..
sum
(k, wd(k)) =e= 1;

ewx(i)..  wx(i) =e=
sum
(j, wgt(i,j));
ewy(j)..  wy(j) =e=
sum
(i, wgt(i,j));
ewd(k)..  wd(k) =e=
sum((i,j)$(ord(i)+ord(j)-1=ord
(k)), wgt(i,j));

compx.. xa =e=
sum
((i,j), xv(i,j)*wgt(i,j));
compy.. ya =e=
sum
((i,j), yv(i,j)*wgt(i,j));
compfv.. fa =e=
sum
((i,j), fv(i,j)*wgt(i,j));


obj..  z =e= YA + 15*XA;

fa.lo = 67;
xa.lo = 227;
xa.up = 229;



model m /all/
;
solve
m minimizing z using mip;