Arrangement of level sets of quadratic constraints and its relation to nonconvex quadratic optimization problems

We study a special class of non-convex quadratic programs subject to two (possibly indefinite) quadratic constraints when the level sets of the constraint functions are {\it not} arranged {\it alternatively.} It is shown in the paper that this class of problems admit strong duality following a tight SDP relaxation, without assuming primal or dual Slater conditions. Our results cover Ye and Zhang’s development in 2003 and the generalized trust region subproblems (GTRS) as special cases. Through the novel geometric view and some simple examples, we can explain why the problem becomes very hard when the level sets of the constraints are indeed arranged alternatively.