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gams:model_an_absolute_value_in_a_linear_model [2008/10/29 12:34]
gams:model_an_absolute_value_in_a_linear_model [2020/05/18 15:29]
Frederik Fiand
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 ===== How do I model an absolute value in a linear model? ===== ===== How do I model an absolute value in a linear model? =====
-You can not put an absolute term for a variable directly ​+You cannot ​put an absolute term for a variable directly ​
 into a linear model. The model fragment below will **not** work: into a linear model. The model fragment below will **not** work:
 <​code>​ <​code>​
-obj..   ​z=e=sum(j, abs(x(j)));+[...] 
 +obj..       ​z=e=sum(j, abs(x(j)));
 cons(i).. ​  ​sum(j,​ a(i,​j)*x(j)) =l= b(i); cons(i).. ​  ​sum(j,​ a(i,​j)*x(j)) =l= b(i);
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 </​code>​ </​code>​
-You have to introduce two positive variables xpos(j) and xneg(j),  +Various error messages will be given:
-and replace:  +
-  * ''​abs(x(j)) =  xplus(j) + xneg(j)''​ +
-  * ''​x(j) = xplus(j) - xneg(j)''​ +
-Thus the correct model fragment ​ is+
 <​code>​ <​code>​
-positive variable ​xplus(j), xneg(j); +  14  solve foo minimizing z using lp; 
-obj..   ​z=e=sum(j, ​xplus(j) + xneg(j)); +****                                 ​$51,​256 
-cons(i).. ​  ​sum(j, a(i,j)*(xplus(j) - xneg(j))) =l= b(i);+****  51  Endogenous function argument(s) not allowed in linear models 
 +**** 256  Error(s) in analyzing solve statement. More detail appears 
 +****      Below the solve statement above 
 +**** The following LP errors were detected in model foo: 
 +****  51 equation obj.. the function ABS is called with non-constant arguments 
 +Instead of using the ''​abs()''​ function, you could introduce two positive variables ''​xpos(j)''​ and ''​xneg(j)''​ and substitute:  
 +  * ''​abs(x(j)) =  xpos(j) + xneg(j)''​ 
 +  * ''​x(j) = xpos(j) - xneg(j)''​ 
 +This reformulation splits the ''​x(j)''​ into a positive part ''​xpos(j)''​ and a negative part ''​xneg(j)''​. Note that this only works if ''​abs(x(j))''​ is minimized, because in that case, either ''​xpos(j)''​ or ''​xneg(j)''​ is forced to zero in an optimal solution. 
 +The reformulated model fragment ​ is:  
 +positive variable ​xpos(j), xneg(j); 
 +obj..        z=e=sum(j, ​xpos(j) + xneg(j)); 
 +cons(i).. ​   sum(j, a(i,j)*(x(j))) =l= b(i); 
 +xsplit(j).. ​ x(j) =e= xpos(j) - xneg(j);  ​
 model foo /all/; model foo /all/;
IMPRESSUM / LEGAL NOTICEPRIVACY POLICY gams/model_an_absolute_value_in_a_linear_model.txt · Last modified: 2020/05/19 07:00 by Frederik Fiand