# Integer Programming Conditional Constraints

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

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.

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.

• That looks correct. You can choose $M=1$ for constraint 1., $M=2$ for constraint 2. and $M=3$ for constraint 3. – Kuifje Nov 22 '15 at 16:44