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===Roof Duality===
If ''f'' is a quadratic polynomial, a concept called ''roof duality'' can be used to obtain a lower bound for its minimum value.<ref name=Boros2002>{{cite journal |last1=Boros |first1=E. |last2=Hammer |first2=P. L. |year=2002 |title=Pseudo-Boolean Optimization |journal=[[Discrete Applied Mathematics]] |doi=10.1016/S0166-218X(01)00341-9 |doi-access=free |volume=123 |issue=1–3 |pages=155–225 |hdl=2268/202427 |url=http://orbi.ulg.ac.be/handle/2268/202427|hdl-access=free }}</ref> Roof duality may also provide a partial assignment of the variables, indicating some of the values of a minimizer to the polynomial. Several different methods of obtaining lower bounds were developed only to later be shown to be equivalent to what is now called roof duality.<ref name=Boros2002/>
===Quadratizations===
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