On Conic Relaxations of Generalization of the Extended Trust Region Subproblem

Abstract

The extended trust region subproblem (ETRS) of minimizing a quadratic objective over the unit ball with additional linear constraints has attracted a lot of attention in the last few years due to its theoretical significance and wide spectra of applications. Several sufficient conditions to guarantee the exactness of its semidefinite programming (SDP) relaxation or second order cone programming (SOCP) relaxation have been recently developed in the literature. In this paper, we consider a generalization of the extended trust region subproblem (GETRS), in which the unit ball constraint in ETRS is replaced by a general, possibly nonconvex, quadratic constraint. We demonstrate that the SDP relaxation can further be reformulated as an SOCP problem under a simultaneous diagonalization condition of the quadratic form. We then explore several sufficient conditions under which the SOCP relaxation of GETRS is exact under Slater condition.