广义信赖域子问题局部非全局极小值点的新结果
New results on the local-nonglobal minimizers of the generalized trust-region subproblem
AI总结：
本文研究广义信赖域子问题及其等式约束版本的局部非全局极小值点，在联合正定或负定条件下证明其数量上界，并给出寻找全部此类极小值点或确认其不存在的算法。
摘要：
本文研究广义信赖域子问题 (GTR) 及其等式约束版本 (GTRE) 的局部非全局极小值点。首先，在联合确定性假设下，建立了 (GTR) 与 (GTRE) 的局部非全局极小值点之间的等价性。同时，给出一个反例，否定了 Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293] 的一个猜想。其次，若 Hessian 矩阵对为联合正定，则证明 (GTR) 和 (GTRE) 各自至多有两个局部非全局极小值点。该结果首次证实了 Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293] 的另一个猜想的正确性。第三，若 Hessian 矩阵对为联合负定，则验证 (GTR) 和 (GTRE) 各自至多有一个局部非全局极小值点。特别地，当约束退化为球或球面时，上述结果正是经典的 Martínez 结果。最后，提出一种算法，用于在给定容差内找出 (GTR) 和 (GTRE) 的所有局部非全局极小值点，或确认其不存在。初步数值结果表明了该算法的有效性。
英文摘要：
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 Martínez'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.
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