# Mixed-integer LP formulation with equality indicator functions in constraints

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.

$$w_i$$ 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 $$\delta_i \in \{0,1\}$$