GAMS — The Solver Manuals - Available Software

CONOPT 125
on the left hand side, and scale factors, both from variables and equations, can be used on the right hand side,
for example to define other scale factors. The default scale factor is always 1, and a scale factor must be positive;
GAMS will generate an execution time error if the scale factor is less than 1.e-20.
The mathematical definition of scale factors is a follows: The scale factor on a variable, V s is used to related the
variable as seen by the modeler, V m , to the variable as seen by the algorithm, V a , as follows:
V m = V a ∗ V s
This means, that if the variable scale, V s , is chosen to represent the order of magnitude of the modeler’s variable,
V m , then the variable seen by the algorithm, V a , will be around 1. The scale factor on an equation, G s , is used
to related the equation as seen by the modeler, G m , to the equation as seen by the algorithm, G a , as follows:
G m = G a ∗ G s
This means, that if the equation scale, G s , is chosen to represent the order of magnitude of the individual terms
in the modelers version of the equation, G m , then the terms seen by the algorithm, G a , will be around 1.
The derivatives in the scaled model seen by the algorithm, i.e. dG a /dV a , are related to the derivatives in the
modelers model, dG m /dV m , through the formula:
dG a /dV a = dG m /dV m ∗ V s /G s
i.e. the modelers derivative is multiplied by the scale factor of the variable and divided by the scale factor of the
equation. Note, that the derivative is unchanged if V s = G s . Therefore, if you have a GAMS equation like
G .. V =E= expression;
and you select G s = V s then the derivative of V will remain 1. If we apply these rules to the example above with
an intermediate variable we can get the following automatic scale calculation, based on an ”average” reference
value for X:
SCALAR XREF; XREF = 6;
Y.SCALE = SUM(P, A(P)*POWER(XREF,ORD(P)-1));
YDEF.SCALE = Y.SCALE;
or we could scale Y using values at the end of the X interval and add safeguards as follows:
Y.SCALE = MAX( ABS(SUM(P, A(P)*POWER(X.LO,ORD(P)-1))),
ABS(SUM(P, A(P)*POWER(X.UP,ORD(P)-1))),
0.01 );
Lower and upper bounds on variables are automatically scaled in the same way as the variable itself. Integer
and binary variables cannot be scaled.
GAMS’ scaling is in most respects hidden for the modeler. The solution values reported back from a solution
algorithm, both primal and dual, are always reported in the user’s notation. The algorithm’s versions of the
equations and variables are only reflected in the derivatives in the equation and column listings in the GAMS
output if OPTION LIMROW and/or LIMCOL are positive, and in debugging output from the solution algorithm,
generated with OPTION SYSOUT = ON. In addition, the numbers in the algorithms iteration log will represent the
scaled model: the infeasibilities and reduced gradients will correspond to the scaled model, and if the objective
variable is scaled, the value of the objective function will be the scaled value.
A final warning about scaling of multidimensional variables is appropriate. Assume variable X(I,J,K) only appears
in the model when the parameter IJK(I,J,K) is nonzero, and assume that CARD(I) = CARD(J) = CARD(K)
= 100 while CARD(IJK) is much smaller than 100**2 = 1.e6. Then you should only scale the variables that appear
in the model, i.e.