Integer Programming Conditional Constraints

Question

I have a set of integer $\{0,1\}$ variables $x_1$,$y_1$,$x_2$,$y_2$,$x_3$,$y_3$,$x_4$,$y_4$

I want a conditional constraint such that if any of the $x$ variables is equal to 1, I want the sum of the subsequent $y$ variables to be $2$.

For example

1. if $x_1==1$ then $y_2+y_3+y_4=2$,
2. if $x_2==1$ then $y_3+y_4=2$
3. if $x_3==1$ then $y_4=2$

The objective function is just the sum of all the variables. There are additional constraints such as: $x_1+x_2+x_3+x_4=2$ and $y_1+y_2+y_3+y_4=2$.

The solution here would be: $1 0 1 0 0 1 0 1$

Answer 1

I have figured out the answer to this. The constraints:

1. if $x_1$==1 then $y_2+y_3+y_4$=2,
2. if $x_2$==1 then $y_3+y_4$=2
3. if $x_3$==1 then $y_4$==2

can also be formulated as:

1. if $x_1$==1 then $y_1$=0,
2. if $x_2$==1 then $y_1+y_2$=0
3. if $x_3$==1 then $y_1+y_2+y_3$==0

Hence the constraints can be written as such:

1. $y_1$<=$M(1-x_1)$
2. $y_1+y_2$<=$M(1-x_2)$
3. $y_1+y_2+y_3$<=$M(1-x_3)$

Where $M$ is a very large integer.

Answer 2

Such constraints are called disjunctive constraints. You can proceed as follows (for your constraint 1.):

$$ y_2+y_3+y_4\le2+(1-x_1)\quad y_2+y_3+y_4\ge2-2(1-x_1), $$

This way, if $x_1=1$, you have

$$ y_2+y_3+y_4\le2 \quad y_2+y_3+y_4\ge2, $$

which is equivalent to $y_2+y_3+y_4=2$. And if $x_1=0$, you have

$$ y_2+y_3+y_4\le2+1=3\quad y_2+y_3+y_4\ge2-2=0, $$

which will always be satisfied since variables are boolean.