LP constraint enconding

Question

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

Answer 1

Your constraint

$\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 $

is equivalent to

$\sum_{i \in A} \alpha_i x_i \le Const + M y$

$\sum_{i \in A} \alpha_i x_i \ge Const - M y$

$ \sum_{i \in A} \alpha_i x_i + \sum_{i \in B}\beta_i x_i \le Const + M (1-y)$

$ \sum_{i \in A} \alpha_i x_i + \sum_{i \in B}\beta_i x_i \ge Const - M (1-y)$

where $y\in\{0,1\}$ and $M$ is a really large ("large enough") positive constant.

Answer 2

The essential problem is how to express the $\beta_i x_i y$ via linear constraints; as $\beta_i$ and $y \in {0,1}$ that is equivalent to expressing $\eta_i := \beta_i \wedge y$ via linear constraints, which is possible in the following way:

$\eta_i \le \beta_i$
$\eta_i \le y$
$0 \le \eta_i$
$\beta_i+y-1 \le \eta_i$

now replace all occurences of $\beta_i x_i y$ with $\eta_i x_i$ and add the above constraints;
The rest should pose no further difficulties.

The general way of modeling Boolean operations via linear constraints is to take the faces of the simplex that is the convex hull of the 3D vectors, whose $x$ and $y$ coordinates correspond to the values of the variables and whose $z$ coordinate coresponds to the result of the operation.
The upper faces are upper bounds and the lower faces are the lower bounds.