Simplified semidefinite and completely positive relaxations

This paper is concerned with completely positive and semidefinite relaxations of quadratic programs with linear constraints and binary variables as presented in Burer [2]. It observes that all constraints of the relaxation associated with linear constraints of the original problem can be accumulated in a single linear constraint without changing the feasible set of either the completely positive or the semidefinite approximation. It also shows that a tightening of the semidefinite relaxation proposed by Burer in [2] is equivalent to the original relaxation. The simplified relaxation derived in this paper can be approached with an interior-point method with O(n3) arithmetic operations per iteration where n is the number of variables of the initial quadratic program.