binary variable zero when the difference between two continuous is zero

[J C]

Hello everyone!

I have the next set of constraints:

1) 0 <= x(t) <= A;

2) x(t) - x(t-1) <= B*yup(t)

3) x(t-1) - x(t) <= B*ydw(t),

where x is a continuous variable, yup, ydw are binary variables, A and B are positive parameters.

The constraints work well while x(t) < A. But when x(t) = A the  constraints 2 and 3 have x(t) = x(t-1) which implies that yup(t) and ydw(t) must be zero, however  the binary variables take values of 1 in some periods. 

I am thinking in establish a new constraint like this x(t) - x(t-1) = z(t) where z should be a continuous variable that measure the difference between x(t) and x(t-1) and use new auxiliary binary variables to define when x(t) - x(t-1) = 0 or x(t) - x(t-1) > 0.

Anyone have some ideias?

Regards

[AMPL Google Group]

Consider constraint (2) in particular. It expresses the following restriction:

(2a)
either yup[t] = 0 and x[t] - x[t-1] <= 0
    or yup[t] = 1 and x[t] - x[t-1] <= B

I guess this is not acceptable for your model, because when x[t] = x[t-1] it allows both yup[t] = 0 and yup[t] = 1. But I am not sure how you want to fix it. How would you change (2a) so that it is the correct restriction for your model?

--
Robert Fourer
am...@googlegroups.com

{#HS:1656502147-106638#}

[J C]

Dear Robert,

Thanks for your response,

The original constraints are 

1) 0 <= x(t) <= A;

2) -B*ydw(t) <= x(t) - x(t-1) <= B*yup(t)

For example:

Of the constraint (2) it seems that:

When x(t-1) = 5 and x(t)=10, yup(t)=1 and ydw=0, because the difference x(t) - x(t-1) is positive. Now  when x(t-1) = 10 and x(t)=5, yup(t)=0 and ydw=1, because the difference x(t) - x(t-1) is negative. Acts only one binary variable at time. When there are differences between x(t) and x(t-1) all is ok in the model, but if for example t-1) = 5 and x(t)=5, any binary variable should be activate, due there is not change, however, in some periods when there is not change the any binary variable seems active. I am thinking that the (2) says when binary variables are 1, but does not say anything about when it is zero. Similar to x(t) - x(t-1) >= 0.0001*yaux.

Regards

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[AMPL Google Group]

Your constraint (2) implies the following:

• When x[t] - x[t-1] > 0, then yup[t] = 1, but ydw[t] can be either 0 or 1.
• When x[t] - x[t-1] < 0, then ydw[t] = 1, but yup[t] can be either 0 or 1.
• When x[t] = x[t-1], yup[t] can be either 0 or 1, and ydw[t] can be either 0 or 1.

To restrict the binary variables more nearly in the way that you describe, you could add constraint (3) yup[t] + ydn[t] = 1. Then (2) and (3) together imply:

• When x[t] - x[t-1] > 0, then yup[t] = 1 and ydw[t] = 0.
• When x[t] - x[t-1] < 0, then ydw[t] = 1 and yup[t] = 0.
• When x[t] = x[t-1], then either yup[t] = 1 and ydw[t] = 0, or else ydw[t] = 1 and yup[t] = 0.

However it is not possible (when using linear constraints) to say that when x[t] = x[t-1], both yup[t] and ydw[t] must be 0.

--
Robert Fourer
am...@googlegroups.com

{#HS:1656502147-106638#}

On Mon, Oct 11, 2021 at 5:31 PM UTC, AMPL Modeling Language <am...@googlegroups.com> wrote:

Dear Robert,

Thanks for your response,

The original constraints are

1) 0 <= x(t) <= A;
2) -B*ydw(t) <= x(t) - x(t-1) <= B*yup(t)

For example:
Of the constraint (2) it seems that:
When x(t-1) = 5 and x(t)=10, yup(t)=1 and ydw=0, because the difference x(t) - x(t-1) is positive. Now when x(t-1) = 10 and x(t)=5, yup(t)=0 and ydw=1, because the difference x(t) - x(t-1) is negative. Acts only one binary variable at time. When there are differences between x(t) and x(t-1) all is ok in the model, but if for example t-1) = 5 and x(t)=5, any binary variable should be activate, due there is not change, however, in some periods when there is not change the any binary variable seems active. I am thinking that the (2) says when binary variables are 1, but does not say anything about when it is zero. Similar to x(t) - x(t-1) >= 0.0001*yaux.

Regards

On Mon, Oct 11, 2021 at 4:01 PM UTC, AMPL Google Group <am...@googlegroups.com> wrote:

Consider constraint (2) in particular. It expresses the following restriction:

(2a)
either yup[t] = 0 and x[t] - x[t-1] <= 0
    or yup[t] = 1 and x[t] - x[t-1] <= B

I guess this is not acceptable for your model, because when x[t] = x[t-1] it allows both yup[t] = 0 and yup[t] = 1. But I am not sure how you want to fix it. How would you change (2a) so that it is the correct restriction for your model?

--
Robert Fourer
am...@googlegroups.com