[AMPL 24819] linearizing conditional constraints

Google Group Mirror
1

Dear team,

I have developed the next constraints for the next conational:

I want a binary variable let say z[t]=0, when 0<= v[t] < VMAX, that means, while v[t] is zero in t, then z[t] = 0, On other hand, z[t]=1, when VMIN <= v[t] <= VMAX. To get that I formulated the next two constraints:

  1. v[t] >= VMIN*z[t]
  2. v[t] <= (VMIN - eps)(1 - z[t]) +VMAXz[t], where eps is a smallest 1E-5 value.

In this case, I want to use the binary variable z[t] the next equation:

r[t] = x[t] + s[t] + H[t], where r, x, y are no negative variables and H[t] is a parameter. Here I want to say that while z[t]=0, that is 0<= v[t] < VMAX, then r[t] = H[t] and x[t] = s[t] = 0, and only after v[t] reached the VMIN value, that is VMIN <= v[t] <= VMAX, then r[t] = x[t] + s[t] + H[t], where x and s are positive different of zero. To get that I made:

r[t] = x[t] + s[t] + H[t]

x[t] <= Mz[t]
s[t] <= Mz[t]

Please help me checking the proposed formulation, and if you have another best option to formulate that, your suggestions always will be important.

Regards

Julio

2

Are you saying that you want z[t] and v[t] to be related like this?

z[t] = 0 ==> v[t] = 0
z[t] = 1 ==> VMIN <= v[t] <= VMAX

That can be written equivalently as these linear constraints:

v[t] >= VMIN * z[t]
v[t] <= VMAX * z[t]

However, the constraints you have written imply that

z[t] = 0 <==> 0 <= v[t] <= VMIN - eps
z[t] = 1 <==> VMIN <= v[t] <= VMAX

If that is what you want, then your constraints are correct. 1e-5 would be a reasonable value to try for eps.

For your other constraints x[t] <= Mz[t] and s[t] <= Mz[t], it should be clear that they only say z[t] = 0 ==> x[t] = 0 and z[t] = 0 ==> s[t] = 0.

In general, if you want to check constraints involving a binary variable, just substitute 0 for the variable and see what constraints you get, then substitute 1 for the variable and see what constraints you get.

3

Thanks, Robert, for your reply.

To understand mare clearly mu problem. I have an equation v[t] = x[t] + s[t] + other variables, and I have other related equation r[t] = x[t] + s[t] + H[t], where r, x, y are no negative variables and H[t] is a parameter, also VMIN <= v[t] <= VMAX, where VMIN is not zero. v[t] represents the water volume of a reservoir. I start in the first-time t0 of analysis with the reservoir empty, so, I want that while 0 <= v[t] < VMIN i.e. While v[t] does not reach the VMIN the variables x[t] = s[t] = 0 and r[t]=H[t] where H[t] is an external inflow.

Summarizing, I want to do that:

z[t] = 0 if 0 <= v[t] < VMIN
z[t] = 1 if VMIN <= v[t] <= VMAX

and

r[t] = x[t] + s[t] + H[t]

x[t] <= Mz[t]
s[t] <= Mz[t]

Would represent the equations in the previous email the above equations.

Regards

Julio

4

Given your description of the constraints that you want to include, your linearization looks correct to me.

5

Thanks dear Robert