positive variables Qpos, Qneg;
Q =e= Qpos - Qneg;
- objextive now becomes 2*(Qpos+Qneg)
I was just editing something in the next generation of GAMS documentation about this very example, so I’ll comment here that you need to be minimizing abs(Q). If you’re maximizing then you’ll have to go to a MIP or perhaps a nonlinear formulation, but the nonlinear problem will be nonconvex so you might as well leave it as a MIP.
Of course Steve, I forgot to mention that in my reply to the question. My bad.
both variables should be defined as positive.
positive variables Qpos, Qneg;
free variables Q, Z “objective variable”;
equation Absolute_definition_of_Q;
Absolute_definition_of_Q… Q = Qpos-Qneg;
equation Objective;
Objective… Z =g= 2*(Qpos+Qneg);
Absolute_definition_of_Q… Q =e= Qpos-Qneg;
If you want to define Qneg as negative variable, you have to rearrange the signs in all the equations so it makes sense. Defining both as positive variables is the common way to do this.
You are making things way to complicated,
Suppose Q takes on value 10
then due to the equation Q =e= Qpos -Qneg;
Qpos takes on value 10 and Qneg takes on value 0;
Suppose Q takes on value -10
then Qpos takes on value 0 and Qneg talkes on value 10;
in either case the objective variable Z because of Z =g= 2*(Qpos + Qneg);
takes on value 20;
This is the bais for least absolute deviations calculations (although you don’t need the factor 2, it is completely redundant).
Yanzhiping,
I am attaching a little example that has 3 versions of the same model. A variant of this will be in the updated docs so I thought I’d give it a trial run here.
The first uses abs() and therefore solves as a DNLP. This is not the recommended way. It is what you want to avoid.
The second reformulates abs() using the trick Gideon mentioned in his first post. The objective is linear, but since we have a nonlinear constraint we solve using NLP. A nice, smooth NLP. This works since the objective is min(abs(x)). This is the way to go.
The third formulation approximates the abs function via a nonlinear smoothing function. You don’t need this in your case so you can ignore this if you find it confusing, but if the objective was working against you rather than for you, this might be useful.
HTH,
-Steve
absReform.gms (1.04 KB)
Hi Steve,
May I suggest changing equation f into:
f … sqr(x-cx) + sqr(-y-cy) =L= 1;
This way you y will be negative and it shows better that the abs tricks work
best,
Gideon
Gideon,
Thanks for the suggestion. Indeed, it helps to show the reform tricks are working when the abs() function is “evaluated” correctly on both branches.
-Steve
Yanzhiping,
Thet way I initially formulated the response to your question is that you have some sort of variable Q that can be either positive or negative. I assumed you wanted to have the absolute value of Q.
If this is not the case I misunderstood your question. If thgis is the case, the code I provided is the correct way to model this problem.
best,
Gideon
but in this way the abs() will make the problem non-linear.
เย็ดสาว
Hi Phallylka,
That’s why you use the Steve and I described, so your system remains linear.
Best, Gideon