Linearize nonquadratic nonlinear constraint

Question

Is there anyway I can linearize the following nonquadratic nonlinear constraints

$$ y_i \geq \frac{2^{\left(\sum_j x_{ij}a_j / C_1 \right)}-1}{C_2}$$

where $x_{ij}$ are binary variables, $y_i$ is continuous variable, and $C_1,C_2,a_j$ are constants

Answer 1

The trivial linearization is to enumerate the cases, since the nonlinearity only involves the binary variables.

Example, if you have $y_1 \geq 2^{x_1+x_2}$, you are equivalently saying that $y_1 \geq 1$ if $x_1 = x_2 =0$, $y_1 \geq 2$ if $x_1 = 1, x_2 =0$, $y_1 \geq 2$ if $x_1 = 0, x_2 =1$, and $y_1 \geq 4$ if $x_1 = 1, x_2 =1$. Those constraints can be written as a set of linear constraints.