The question is how to arrive at a system in which another equation is enforced based on the value of a function. To be fairly general, use the following statement:
z = f(x) when y > 0 z = g(x) when y < 0
Note that the distinction between
>= and just
> is not meaningful for a numerical algorithm on a finite precision machine.
Start by writing, in GAMS,
Variable fs, gs; z =E= (f(x) - fs) + (g(x) - gs);
y > 0, we want
fs = 0 and
gs = g(x). When
y < 0, we want
fs = f(x) and
gs = 0;
Now declare a binary variable,
b, that will be 1 when
y > 0 and 0 and
y < 0.
Binary Variable b; Positive Variable yp, yn; y =E= yp - yn; yp =L= ymax * b; yn =L= ymax * (1 - b);
Next, split the two terms of
z into positive and negative parts.
Positive Variable fp, fn, gp, gn; f(x) - fs =E= fp - fn; g(x) - gs =E= gp -gn; fp + fn =L= fmax * b; gp + gn =L= gmax * (1 - b);
b = 0 ⇒ fs = f(x) and
b = 1 ⇒ gs = g(x).
Finally split just the f and g slacks fs and gs into positive and negative components.
Positive Variable fsp, fsn, gsp, gsn; fs =E= fsp - fsn; gs =E= gsp - gsn; fsp + fsn =L= fmax* (1 - b); gsp + gsn =L= gmax * b;
b= 0 ⇒ gs = 0; and b = 1 ⇒ fs = 0;
Taken together, the last two sections give
b = 0 ⇒ z = g(x) and
b = 1 ⇒ z = f(x).