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 K ⊆Rn with additional linear and binary constraints can be rewritten as 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 a results obtained by Sturm and Zhang ([J.F. Sturm and S. Zhang, On cones of nonnegative quadratic functions. Math. Oper. Res. 28 (2003)]), since we can handle problems with many linear and binary constraints. Our result also generalizes the well-known completely positive representation result from Burer ([S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120 (Ser. A) (2009), pp. 479-495]), which is actually a special instance of our result with K = R+.