gams:use_different_equations_based_on_the_value_of_a_function

This shows you the differences between two versions of the page.

gams:use_different_equations_based_on_the_value_of_a_function [2007/08/10 10:56] |
gams:use_different_equations_based_on_the_value_of_a_function [2007/08/10 10:56] (current) |
||
---|---|---|---|

Line 1: | Line 1: | ||

+ | ====== Using different equations based on the value of a function ====== | ||

+ | The question is how to have to regimes where a different equation is enforced based on the value of some function. To be fairly general, use the following statement: | ||

+ | <code> | ||

+ | z = f(x) when y > 0 | ||

+ | z = g(x) when y < 0 | ||

+ | </code> | ||

+ | Note that the distinction between ''>='' and just ''>'' is not meaningful for a numerical algorithm on a finite precision machine. | ||

+ | |||

+ | Start by writing, in GAMS, | ||

+ | <code> | ||

+ | Variable fs, gs; | ||

+ | z =E= (f(x) - fs) + (g(x) - gs); | ||

+ | </code> | ||

+ | 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''. | ||

+ | <code> | ||

+ | Binary Variable b; | ||

+ | Positive Variable yp, yn; | ||

+ | y =E= yp - yn; | ||

+ | |||

+ | yp =L= ymax * b; | ||

+ | yn =L= ymax * (1 - b); | ||

+ | </code> | ||

+ | Next, spit the two terms of ''z'' into positive and negative parts. | ||

+ | <code> | ||

+ | 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); | ||

+ | </code> | ||

+ | 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. | ||

+ | <code> | ||

+ | 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; | ||

+ | </code> | ||

+ | 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)''. | ||

IMPRESSUM / LEGAL NOTICE
PRIVACY POLICY
gams/use_different_equations_based_on_the_value_of_a_function.txt ยท Last modified: 2007/08/10 10:56 (external edit)