New results on the local-nonglobal minimizers of the generalized trust-region subproblem

In this paper, we study the local-nonglobal minimizers of the Generalized Trust-Region subproblem $(GTR)$ and its Equality-constrained version $(GTRE)$. Firstly, the equivalence is established between the local-nonglobal minimizers of both $(GTR)$ and $(GTRE)$ under assumption of the joint definiteness. By the way, a counterexample is presented to disprove a conjecture of Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293]. Secondly, if the Hessian matrix pair is jointly positive definite, it is proved that each of $(GTR)$ and $(GTRE)$ has at most two local-nonglobal minimizers. This result first confirms the correctness of another conjecture of Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293]. Thirdly, if the Hessian matrix pair is jointly negative definite, it is verified that each of $(GTR)$ and $(GTRE)$ has at most one local-nonglobal minimizer. In special, if the constraint is reduced to be a ball or a sphere, the above result is just the classical Mart\'{i}nez's. Finally, an algorithm is proposed to find all the local-nonglobal minimizers of $(GTR)$ and $(GTRE)$ or confirm their nonexistence in a tolerance. Preliminary numerical results demonstrate the effectiveness of the algorithm.