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);
```

So, when *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);
```

So, *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;
```

So, *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)*.