Burer’s key assumption for semidefinite and doubly nonnegative relaxations

Burer has shown that completely positive relaxations of nonconvex quadratic programs with nonnegative and binary variables are exact when the binary variables satisfy a so‐called key assumption. Here we show that introducing binary slack variables to obtain an equivalent problem that satisfies the key assumption will not improve the semidefinite relaxation. In contrast, such slack variables will improve the doubly nonnegative relaxation, but the same improvement can be obtained in a simpler fashion by adding certain linear inequality constraints.