Globally solving the trust region subproblem using simple first-order methods

We consider the trust region subproblem which is given by a minimization of aquadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known secondorder necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions correspondsto one of two possible scenarios that occur in this problem, commonly referred to in the literatureas the presence or absence of the\hard case". We consider a family of first-order methods, whichincludes the projected and conditional gradient methods. We show that any method belonging tothis family produces a sequence which is guaranteed to converge to a stationary point of the trustregion subproblem. Based on this result and the established sufficient optimality conditions, we showthat convergence to an optimal solution can be also guaranteed as long as the method is properlyinitialized. In particular, if the method is initialized with the zeros vector and reinitialized with arandomly generated feasible point, then the best of the two obtained vectors is an optimal solutionof the problem with probability 1.