I have an objective function to be maximized

$obj(x) = \sum_i \gamma_i x_i$ with $x_i \in \mathbb{R}$

With multiple constraints of the form:

$\min_{y \in 0,1} (\sum_{i \in A} \alpha_i x_i + \sum_{i \in B}\beta_i x_i y )= Const$, with $\alpha_i,\beta_i \in {0,1}$ and $A$ and $B$ disjoint

Each of such constraints should probably be reformulated as:

$\sum_{i \in A} \alpha_i x_i \geq Const$

$\sum_{i \in A} \alpha_i x_i + \sum_{i \in B}\beta_i x_i \geq Const$

$\sum_{i \in A} \alpha_i x_i = Const \quad OR \quad \sum_{i \in A} \alpha_i x_i + \sum_{i \in B}\beta_i x_i = Const $

But I am not sure on how to encode this OR condition as a linear constraint.

Any ideas?

EDITS:

Made clear that there are multiple of such constraints between, up to 20.

Reformulated the problem more precisely

all$x_0+\dots=Const$ and $x_0\ge Const$. So you just solve 2 standard LP problems instead of one. That is if the constant is the same. If not, split according to the value of Const: $x_0$ can be just one of them, if any. $\endgroup$