  This paper presents global optimal solutions to a nonconvex quadratic
minimization problem over a sphere constraint. The problem is well-known as a
trust region subproblem and has been studied extensively for decades. The main
challenge is the so called 'hard case', i.e., the problem has multiple
solutions on the boundary of the sphere. By canonical duality theory, this
challenging problem is able to reformed as an one-dimensional canonical dual
problem without duality gap. Sufficient and necessary conditions are obtained
by the triality theory, which can be used to identify whether the problem is
hard case or not. A perturbation method and the associated algorithms are
proposed to solve this hard case problem. Theoretical results and methods are
verified by large-size examples.
