On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets

In the paper we prove that any nonconvex quadratic problem over some set with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246-267, 2003). Our result also generalizes the well-known completely positive representation result from Burer (Math Program 120:479-495, 2009), which is actually a special instance of our result with K = R-+(n).