Title: Extended Trust-Region Problems with One or Two Balls: Exact Copositive and Lagrangian Relaxations,
Abstract: We establish a geometric condition guaranteeing exact copositive relaxation
for the nonconvex quadratic optimization problem under two quadratic and
several linear constraints, and present sufficient conditions for global
optimality in terms of generalized Karush-Kuhn-Tucker multipliers. The
copositive relaxation is tighter than the usual Lagrangian relaxation. We
illustrate this by providing a whole class of quadratic optimization problems
that enjoys exactness of copositive relaxation while the usual Lagrangian
duality gap is infinite. Finally, we also provide verifiable conditions under
which both the usual Lagrangian relaxation and the copositive relaxation are
exact for an extended CDT (two-ball trust-region) problem. Importantly, the
sufficient conditions can be verified by solving linear optimization problems.