# How to convert a non-linear constraint to a linear constraint for integer programming?

I have non-linear scheduling model and I want to convert it to a linear model. But I have no idea about how can I do it.

The nonlinear constraint is:

For each $i, i'\in I$ and $j, j' \in J$ and $q, q' \in Q$ and $k \in K$

$$s_{ij} \geq (\sum_{q'=q}^5 o_{i'j'q'} * c_{i'j'} * x_{i'j'k}) * x_{ijk}*o_{ijq}$$

In this constraint:

$s$ is continuous positive variable,

$o$ and $x$ are binary variables

I can do the $x_{ijk}*o_{ijq}$ part using a Big M number but I don't know any about the summation inside the paranthesis. How can I do it? Please help.

Expand the summation. You will get something like $\sum c \cdot o \cdot x \cdot x \cdot o$. This is constant * binary * binary * binary * binary. The multiplication of 4 binary variables $y=x_1x_2x_3x_4$ can be linearized as: \begin{align} &y \le x_i \\ &y \ge x_1+x_2+x_3+x_4-3 \\ &y \in \{0,1\} \>\text{(note: $y$ can even be relaxed to continuous)} \end{align} No big-Ms needed I think.