gams:use_different_equations_based_on_the_value_of_a_function

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);

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)`

.

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gams/use_different_equations_based_on_the_value_of_a_function.txt · Last modified: 2020/05/26 12:57 by Lutz Westermann