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Is there a way to formulate the following Linear Program in a mixed-integer LP with big-M modelling?

$\max_{w_{i}}\sum_{i=1}^{N}w_{i}\cdot C_{i}$

subject to:

(1) $I\left(w_{i}=0\right)K_{2}+w_{i}\geq K_{2},\forall i$

where

$I\left(\cdot\right)=\begin{cases} 1 & \text{if } w_i =0\\ 0 & \text{otherwise} \end{cases}$

$K_1, K_2, C_i$ are just constants, and the idea is to have $w_i$ either bigger than $K_2$ or $0$.

The difference with other questions is that there is an equality in the indicator function.

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$w_i$ larger than $K_2$ or $0$ is called semi-continuous, and is a logic supported natively in some solvers. To model it manually with a big-M strategy, you would do $-M\delta_i \leq w_i \leq M\delta_i, w_i \geq K_2\delta_i$, with $\delta_i \in \{0,1\}$

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  • $\begingroup$ Thanks so much Johan! I just edited to precise that delta is binary! $\endgroup$ – Adrien A. Jul 4 at 20:03

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