Exact SDP Relaxations for a Class of Quadratic Programs with Finite and Infinite Quadratic Constraints

Abstract

Abstract. We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality constraints (semi-infinite QCQPs). Sufficient conditions for the exactness of SDP relaxations for QCQPs with finitely many constraints have been extensively studied, notably by Argue, Kilinç-Karzan, and Wang [ Math. Oper. Res., 48 (2023), pp. 100–126], Arima, Kim, and Kojima [ SIAM J. Optim., 34 (2024), pp. 3194–3211], and Joyce and Yang [ Math. Program., 205 (2024), pp. 539–558]. In this work, we present three new sufficient conditions that generalize the existing conditions in these works for both finite and semi-infinite QCQPs. Specifically, we establish relationships among the proposed and existing conditions, and prove that one of the proposed conditions is the weakest among them, since it is implied by all the others. Illustrative examples are also provided to demonstrate the effectiveness of the proposed conditions in comparison to the existing ones.