# 0,1 binary polynomial programming

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)...

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If the $s_{i}$ are integers, there are reformulations of integer polynomial terms that result in mixed-integer (linear) programs, at the cost of introducing additional variables and constraints. Fred Glover has a sequence of papers to that effect in the mid-to-late 1970s, and subsequent work has built upon it.