在 MILP/MIP 中建模 if-then-else-逻辑

Question

I would like to model the following for a mixed-integer linear programming problem : Let y be a binary and x1 and x2 be continuous variables, whereas k1 is an invariant parameter.

if y == 1 then:  
    x2 = k1*x1   
else (y == 0):  
    x2 = 0

First idea that comes to mind is to do something like:

x2 >= k1*x1 - M*(1-y)  
x2 <= k1*x1 - M*(1-y)

But here M would have to be k1*x1 and is therefore not an invariant parameter anymore. Does anyone have a better idea. Thank you!

Answer 1

Per your problem: y: binary_variable x1: continous_variable x2: continous_variable

Please see equations below:

x2 <= u*y
x2 <= k1*x1
x2 >= k1*x1 − u*(1 − y)
x2 >= 0

u is the upper bound on k1*x1
0 <= k1*x1 <= u

you don't require a big M over here.

when y == 1 => x2 == k1*x1
when y == 0 => x2 == 0