I have learned a sigmoid function $\sigma(\theta^Tx)$ using logistic regression. Where, $\theta \in R_+^n$ are the weights and $x \in R_+^n$ is a feature vector. How can I incorporate a linear or quadratic approximation of the learned sigmoid function $\sigma(\theta^Tx)$ as a constraint in my optimization problem.
$$\max_{\alpha \in \{0,1\} \\ x \in R_+^n} F(x,\alpha)\\ s.t. \alpha = \sigma(\theta^Tx)$$
PS: I am okay if the solution requires to first replace the sigmoid function with a Heaviside function $H(\theta^Tx)$ and then do the linear/quadratic approximation.