I have a minimization problem minimizing $d_k \geq 0$ and some other variables with all strictly positive coefficients. I leave my objective function below to better convey my goal.

$$\min_{\mathbf{d},\mathbf{y^+},\mathbf{y^-}} \sum_{k\in\mathcal{K}} C_k d_k +\sum_{(i,j)\in\mathcal{E},k\in\mathcal{K}}P\lambda_kT_{ij}\left(y_{ijk}^+ +y_{ijk}^-\right)$$

To determine the value of $d_k$, I have the following constraint which can possibly become undefined due to the domains of variables associated. Simply, if all $y^+$ and $y^-$ variables are 0...

$$d_k \geq \frac{\displaystyle\sum_{i\in \mathcal{I}_k}\lambda_k y_{ikk}^+}{\displaystyle\sum_{(i,j)\in\mathcal{E}} T_{ij}\left(y_{ijk}^+ +y_{ijk}^-\right)}\qquad \forall k\in \mathcal{K}$$

Notation Domains:

$$\begin{array}{c|c|c} & \text{Domain} & \text{Type}\\ \hline d_k & \mathbb{R}^{\geq0} & \text{Variable} \\ \hline y_{ijk}^+ & \{0,1\} & \text{Variable} \\ \hline y_{ijk}^- & \{0,1\} & \text{Variable} \\ \hline \lambda_k & \mathbb{R}^{>0} & \text{Parameter} \\ \hline T_{ij} & \mathbb{R}^{>0}& \text{Parameter} \\ \hline C_k & \mathbb{R}^{>0}& \text{Parameter} \\ \hline P & \mathbb{R}^{>0}& \text{Parameter} \\ \end{array}$$


The sets: $\mathcal{I}$, $\mathcal{K}$, $\mathcal{E}$ are some polynomial size sets and $\mathcal{I}_k$ is a subset.

Is there a way to deal with nonlinearity and "0/0"?


Multiply both sides of your $d_k \ge$ constraint by the denominator and then linearize $d_k y_{ijk}^+$ and $d_k y_{ijk}^-$ as described in this thread.

  • $\begingroup$ Well, this was sort of in my mind for a while and I do not know why I forgot about it once I posted this question. Anyways, thanks for the response! $\endgroup$ – Taner Cokyasar Feb 20 '20 at 17:02

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