本文提出了求解无约束优化问题的改进$q$-信赖域算法。该算法具有新的信赖域半径更新规则。在函数连续$q$-可微等条件下,我们建立了求解无约束优化问题的改进$q$-信赖域算法的收敛性。最后,数值实验表明,我们的算法是有效的。与Zhou提出的改进信赖域算法相比,我们提出的改进$q$-信赖域算法不仅可以更快地迭代到最优点,还可以解决具有多个最优点的优化问题。本文所得的结果推广和改进了文献中的一些已有结果。
In this paper, we propose an improved $q$-trust region algorithm for solving unconstrained optimization problems. The algorithm has a new rule for updating the radius of the trust region. We establish the convergence of the improved $q$-trust region algorithm for solving unconstrained optimization problems under the conditions that the function is continuously $q$-differentiable and so on. Finally, numerical experiments show that our algorithm is effective. Compared with the improved $q$-trust region algorithm proposed by Zhou, our proposed improved $q$-trust region algorithm not only iterates to the optimal point faster, but also solves optimization problems with multiple optimal solutions. The results obtained in this paper extend and improve some existing results in the literature.
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