Closing the Gap between Necessary and Sufficient Conditions for Local Nonglobal Minimizer of Trust Region Subproblem

Jiulin Wang, Yong Xia

Abstract: The trust region subproblem has at most one local nonglobal minimizer. In characterizing this local solution, there is a clear gap between necessary and sufficient conditions. In this paper, we surprisingly show that the sufficient second-order optimality condition remains necessary. As an application, we improve the state-of-the-art algorithm for computing a candidate of the local nonglobal minimizer and then show that finding the local nonglobal minimizer or proving the nonexistence can be done in polynomial time.