I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to do it?
For example:
I want to convert ${ x }_{ 1 }{ x }_{ 2 }{ x }_{ 3 }$ into quadratic unconstrained term:
$\sum _{ i,j\quad \in \quad poly }^{ \quad }{ { J }_{ij}{ x }_{ i }{ x }_{ j } } $
In order to do this, in my understanding, I should introduce penalty terms to force auxiliary variable build connections for quadratic terms, some thing like:
Corresponding quadratic term:
$f\left( x \right) =\sum _{ i,j\quad \in \quad poly }^{ \quad }{ { J }_{ij}{ x }_{ i }{ x }_{ j } } +\gimel \overrightarrow { PE } $
$\overrightarrow { PE } $ is constrains vector, $\gimel$ is penalty factor.
I should calculate $\overrightarrow { PE } $ and $\gimel$.
Is my idea correct? Can anyone give me some advices of how to reduce degree of a polynomial?
Thanks