Modeling the absolute value
Modeling the absolute value
Hi guys,
I just started to use GAMS (so I have no experience whatsoever). I wanted to know whether and how it is possible to model the absolute values of binary variables. So I want to maximize the difference:
max: SUM(t=0 to T) {abs(x_t  c_t)}
where x_t is a binary decision variable and c_t is also a binary variable (but not decision variable; the values are fixed). Of course there are contraints such that just flipping the value of c_t for every time slot is not feasible.
Does anyone have an idea whether something like this is possible or not.
I just started to use GAMS (so I have no experience whatsoever). I wanted to know whether and how it is possible to model the absolute values of binary variables. So I want to maximize the difference:
max: SUM(t=0 to T) {abs(x_t  c_t)}
where x_t is a binary decision variable and c_t is also a binary variable (but not decision variable; the values are fixed). Of course there are contraints such that just flipping the value of c_t for every time slot is not feasible.
Does anyone have an idea whether something like this is possible or not.

 User
 Posts: 45
 Joined: 11 months ago
 Location: Rosario  Argentina
Re: Modeling the absolute value
Hi, maybe you can try with this restriction:
absvalue(t) correspond to the absolute value of (x(t)  c(t))
Good luck
Code: Select all
eq1(t).. 1  x(t) + c(t) + absvalue(t) =g= 1;
eq2(t).. 1  x(t) + 1c(t) + 1absvalue(t) =g= 1;
eq3(t).. x(t) + 1c(t) + absvalue(t) =g= 1;
eq4(t).. x(t) + c(t) + 1 absvalue(t) =g= 1;
eq5.. z =e= sum(t,absvalue(t) )
Good luck
Re: Modeling the absolute value
Sorry it has been a while since I posted this question. Nonetheless I unforunately have not solved this problem. Basically my problem has slightly changend. Now I want to solve the following prolbem:
max: SUM(t=0 to T) {abs(x_t  y_t)}
where x_t is a binary decision variable and y_t is also a binary decision variable.
Is there a way how this can automatically be done by GAMS meaning that GAMS for example linearizes the abs function?
max: SUM(t=0 to T) {abs(x_t  y_t)}
where x_t is a binary decision variable and y_t is also a binary decision variable.
Is there a way how this can automatically be done by GAMS meaning that GAMS for example linearizes the abs function?
Re: Modeling the absolute value
Hi, Sorry that I have not written anything since a fairly long time. But I do not really understand the approach form Manassaldi and my objective has slightly changed. So I want to maximize the difference:
max: SUM(t=0 to T) {abs(x_t  y_t)}
where x_t is a binary decision variable and y_t is also a binary decision variable.
I heard that something like this is solable with the Gurobi Solver by introducing an auxiliary variable z(t) = x(t)  y(t) for each t. However adding the following equations to GAMS:
led to the error message: "Endogenous function argument(s) not allowed in linear models"
Does anyone have an idea whether and how something like this is possible or not by using GAMS?
max: SUM(t=0 to T) {abs(x_t  y_t)}
where x_t is a binary decision variable and y_t is also a binary decision variable.
I heard that something like this is solable with the Gurobi Solver by introducing an auxiliary variable z(t) = x(t)  y(t) for each t. However adding the following equations to GAMS:
Code: Select all
eq_z(t).. z(t) =e= x(t)  y(t);
eq_z_abs(t).. z_abs(t)=e= abs(z(t));
Does anyone have an idea whether and how something like this is possible or not by using GAMS?

 User
 Posts: 45
 Joined: 11 months ago
 Location: Rosario  Argentina
Re: Modeling the absolute value
Hi!
These constraints are obtained using logical propositions and the socalled "basic steps".
You can not use the abs() function in linear programming
If you observe the restrictions you can note that:
If x = 1 and c = 0 then for eq1 absvalue = 1 (equal to abs(xc)) (the remaining restrictions are also satisfied)
If x = 1 and c = 1 then for eq2 absvalue = 0 (equal to abs(xc)) (the remaining restrictions are also satisfied)
If x = 0 and c = 1 then for eq3 absvalue = 1 (equal to abs(xc)) (the remaining restrictions are also satisfied)
If x = 0 and c = 0 then for eq4 absvalue = 0 (equal to abs(xc)) (the remaining restrictions are also satisfied)
As the variables are binary, there are no other possibilities
Best regard
These constraints are obtained using logical propositions and the socalled "basic steps".
You can not use the abs() function in linear programming
Code: Select all
eq1(t).. 1  x(t) + c(t) + absvalue(t) =g= 1;
eq2(t).. 1  x(t) + 1c(t) + 1absvalue(t) =g= 1;
eq3(t).. x(t) + 1c(t) + absvalue(t) =g= 1;
eq4(t).. x(t) + c(t) + 1 absvalue(t) =g= 1;
eq5.. z =e= sum(t,absvalue(t) )
If x = 1 and c = 0 then for eq1 absvalue = 1 (equal to abs(xc)) (the remaining restrictions are also satisfied)
If x = 1 and c = 1 then for eq2 absvalue = 0 (equal to abs(xc)) (the remaining restrictions are also satisfied)
If x = 0 and c = 1 then for eq3 absvalue = 1 (equal to abs(xc)) (the remaining restrictions are also satisfied)
If x = 0 and c = 0 then for eq4 absvalue = 0 (equal to abs(xc)) (the remaining restrictions are also satisfied)
As the variables are binary, there are no other possibilities
Best regard
Re: Modeling the absolute value
Hi,
I inserted following equations:
z(t) is an auxilliary variable
I think that this might be same as in your example, doesn't it?
I inserted following equations:
Code: Select all
eq_z_firstCondition(t).. z(t) =l= 2  x(t)  y(t);
eq_z_secondCondition(t).. z(t) =l= x (t) + y(t);
...
u
=e=
sum((t), z(t));
...
Solve model using mip maximizing u;
I think that this might be same as in your example, doesn't it?

 User
 Posts: 45
 Joined: 11 months ago
 Location: Rosario  Argentina
Re: Modeling the absolute value
Hi, your equations are good too.
If I am not mistaken, these restrictions are only valid for a maximization problem.
if x=1 y=1
eq_z_firstCondition(t).. z(t) =l= 0;
eq_z_secondCondition(t).. z(t) =l= 2;
so: z=0
if x=1 y=0
eq_z_firstCondition(t).. z(t) =l= 1;
eq_z_secondCondition(t).. z(t) =l= 1;
so: z=1 or z=0 (but for a maximization problem z=1)
if x=0 y=0
eq_z_firstCondition(t).. z(t) =l= 2;
eq_z_secondCondition(t).. z(t) =l= 0;
so: z=0
if x=0 y=1
eq_z_firstCondition(t).. z(t) =l= 1;
eq_z_secondCondition(t).. z(t) =l= 1;
so: z=1 or z=0 (but for a maximization problem z=1)
I think that the socalled "Basic Steps" allow to obtain logical restrictions in a systematic way in a short time.
Best regard!
If I am not mistaken, these restrictions are only valid for a maximization problem.
if x=1 y=1
eq_z_firstCondition(t).. z(t) =l= 0;
eq_z_secondCondition(t).. z(t) =l= 2;
so: z=0
if x=1 y=0
eq_z_firstCondition(t).. z(t) =l= 1;
eq_z_secondCondition(t).. z(t) =l= 1;
so: z=1 or z=0 (but for a maximization problem z=1)
if x=0 y=0
eq_z_firstCondition(t).. z(t) =l= 2;
eq_z_secondCondition(t).. z(t) =l= 0;
so: z=0
if x=0 y=1
eq_z_firstCondition(t).. z(t) =l= 1;
eq_z_secondCondition(t).. z(t) =l= 1;
so: z=1 or z=0 (but for a maximization problem z=1)
I think that the socalled "Basic Steps" allow to obtain logical restrictions in a systematic way in a short time.
Best regard!
Re: Modeling the absolute value
Thanks Manassaldi for your answers and comments.
Would you mind telling me a little bit more about the "basic step". Is that a generall principle for modelling absolut values or what exactly does this method aim for?
Would you mind telling me a little bit more about the "basic step". Is that a generall principle for modelling absolut values or what exactly does this method aim for?

 User
 Posts: 45
 Joined: 11 months ago
 Location: Rosario  Argentina
Re: Modeling the absolute value
Hi, the "basicsteps" are steps to transform logical propositions into algebraic equations.
In your problem, i used 4 logical proposition and three booelan variable:
X: boolean variable equivalent to x
Y: boolean variable equivalent to y
Z: boolean variable equivalent to z (abs(xy))
logical proposition #1: X ^ Y > ¬Z (in logic languaje: X true and Y true implies Z not true)
Now, using the basic steps you can transform the logical proposition #1 into an algebraic equation.
BASIC STEPS
Goal is to Convert Logical Expression into Conjunctive Normal Form (CNF)
1. REPLACE IMPLICATION BY DISJUNCTION
¬(X ^ Y)˅¬Z
2. MOVE NEGATION INWARD APPLYING DE MORGAN’S THEOREM
¬X ˅ ¬Y ˅¬Z
3. RECURSIVELY DISTRIBUTE OR OVER AND
(not necessary in this case)
Finally,
¬X ˅ ¬Y ˅¬Z (CNF)
So, you can replace any CNF for a summatory of each term.
Because at least one of a term has to be true, the sumatory has to be greater than 1
¬X equivalent to 1x
¬Y equivalent to 1y
¬Z equivalent to 1z
So, this CNF generate equation "eq2" of my example
1  x + 1  y + 1  z =g= 1;
logical proposition #2: X ^ ¬Y > Z (generate eq1)
logical proposition #2: ¬X ^ ¬Y > ¬Z (generate eq4)
logical proposition #2: ¬X ^ Y > Z (generate eq3)
This is a quick explanation, I hope it will be useful
Best regard!
In your problem, i used 4 logical proposition and three booelan variable:
X: boolean variable equivalent to x
Y: boolean variable equivalent to y
Z: boolean variable equivalent to z (abs(xy))
logical proposition #1: X ^ Y > ¬Z (in logic languaje: X true and Y true implies Z not true)
Now, using the basic steps you can transform the logical proposition #1 into an algebraic equation.
BASIC STEPS
Goal is to Convert Logical Expression into Conjunctive Normal Form (CNF)
1. REPLACE IMPLICATION BY DISJUNCTION
¬(X ^ Y)˅¬Z
2. MOVE NEGATION INWARD APPLYING DE MORGAN’S THEOREM
¬X ˅ ¬Y ˅¬Z
3. RECURSIVELY DISTRIBUTE OR OVER AND
(not necessary in this case)
Finally,
¬X ˅ ¬Y ˅¬Z (CNF)
So, you can replace any CNF for a summatory of each term.
Because at least one of a term has to be true, the sumatory has to be greater than 1
¬X equivalent to 1x
¬Y equivalent to 1y
¬Z equivalent to 1z
So, this CNF generate equation "eq2" of my example
1  x + 1  y + 1  z =g= 1;
logical proposition #2: X ^ ¬Y > Z (generate eq1)
logical proposition #2: ¬X ^ ¬Y > ¬Z (generate eq4)
logical proposition #2: ¬X ^ Y > Z (generate eq3)
This is a quick explanation, I hope it will be useful
Best regard!
Re: Modeling the absolute value
Thanks again for the answers. Where can I read more about those basic steps? I typed it into google but somehow I could not find useful information about them.