0
$\begingroup$

I have the following optimization problem.

\begin{align} \text{Maximize } & \sum_{k=1}^P x_{i,k} \mu_{k} \\ \text{Subject to } & \sum_{k=1}^P\sum_{l=1}^P x_{i,k}x_{i,l} \sigma_{k, l} \geq \epsilon_i \\ & \sum_{k=1}^P\sum_{l=1}^P x_{i,k}x_{j,l}^*\sigma_{k, l} \leq \gamma_{i, j} \; \forall \; 1 \leq j \leq i - 1 \\ & \sum_{k=1}^P x_{i, k} = K \end{align}

With decision variables $x_{i, k} \in \{0, 1\} \; \forall k \in \{1, 2, \dots, P\}$ and all other variables constant.

My problem is that the first constraint is not linear. In the paper which this program is adapted from (page 12, eq 2.14), they state "We can introduce auxiliary variables to make this formulation linear as is standard in many integer programming formulations.", but I don't know enough about integer linear programming to know how to do that.

How do I make this constraint linear?

$\endgroup$
1
$\begingroup$

Let me introduce variable $y_{i,k,l}= x_{i,k}x_{i,l}$,

We want $$y_{i,k,l} = \begin{cases} 1, & x_{i,k}=1 \text{ and } x_{i,l}=1\\ 0, & \text{Otherwise} \end{cases}$$

Let's replace the first constraint by:

$$y_{i,k,l} \geq x_{i,k} + x_{i,l}-1$$

$$y_{i,k,l} \leq x_{i,k}$$

$$y_{i,k,l} \leq x_{i,l}$$

$$0 \leq y_{i,k,l} \leq 1$$

$$\sum_{k=1}^P\sum_{l=1}^P y_{i,k,l} \sigma_{k, l} \geq \epsilon_i$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.