Largest small polygon

    {  Polygon.mpl   }

    {  GAMS Model Library, http://www.gams.com/modlib/libhtml/polygon.htm  }

    {  Largest small polygon,  NLP,  Size: 54x20  }



TITLE
    Polygon;

OPTIONS
    ModelType=Nonlinear
    ParserType=Extended
Conopt.SDEvalUse = 1;

INDEX
    i := 1..10;
    j := i;

DATA
    RInit[i] := 4*ord(i)*(count(i)+1-ord(i))/(count(i)+1)^2;
    ThetaInit[i] := pi*ord(i)/count(i);

VARIABLES
    R[i]     INITIAL RInit;
    Theta[i] INITIAL ThetaInit;

MODEL

    MAX Polygon_Area = 0.5 * SUM(i<last(i): R[i+1]*R*SIN(Theta[i+1]-Theta));

SUBJECT TO

    Ordered[i<last(i)]:  Theta <= Theta[i+1];

    Distance[i,j] WHERE (j > i):
         R^2 + R[i:=j]^2 - 2*R*R[i:=j]*COS(Theta[i:=j] - Theta) <= 1;

BOUNDS
    0 <= R <= 1;
    0 <= Theta <= Pi;
    R[i=10] = 0;
    Theta[i=10] = Pi;

END