Is there a mathematical optimization branch that explicitly tries to optimize this (type) problem?
$$\eqalign{ & \min \cr & \sum\limits_{i = 1}^N {(J*s[i] + {J_1}*s[i]*s[i + 1] + {J_2}s[i]*s[i + 1]*s[i + 2] + ... + {J_m}s[i]*s[i + 1]*...s[i + m])} \cr & s[i] \in \{ 0,1\} \quad \forall i \in \{ 1,2...,N + m\} \cr} $$
Currently, I only know things about Mixed integer programming (and its implementation using Gurobi)...