In this paper, we propose a normalized Goldstein-type local minimax method (NG-LMM) to seek for multiple minimax-type solutions. Inspired by the classical Goldstein line search rule in the optimization theory in $\mathbb{R}^m$, which is aimed to guarantee the global convergence of some descent algorithms, we introduce a normalized Goldstein-type search rule and combine it with the local minimax method to be suitable for finding multiple unstable solutions of semilinear elliptic PDEs both in numerical implementation and theoretical analysis. Compared with the normalized Armijo-type local minimax method (NA-LMM), which was first introduced in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3):865–885, 2002] and then modified in [Z.Q. Xie, Y.J. Yuan, and J. Zhou, SIAM J. Sci. Comput., 34(1):A395–A420, 2012], our approach can prevent the step-size from being too small automatically and then ensure that the iterations make reasonable progress by taking full advantage of two inequalities. The feasibility of the NG-LMM is verified strictly. Further, the global convergence of the NG-LMM is proven rigorously under a weak assumption that the peak selection is only continuous. Finally, it is implemented to solve several typical semilinear elliptic boundary value problems on square or dumbbell domains for multiple unstable solutions and the numerical results indicate that this approach performs well.

中文翻译：

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寻找半线性椭圆型偏微分方程多重不稳定解的归一化Goldstein型局部极大极小法

在本文中，我们提出了一种归一化的Goldstein型局部最小极大值方法（NG-LMM），以寻求多个最小极大值类型的解。出于优化理论中$ \ mathbb {R} ^ m $的经典Goldstein线搜索规则的启发，该规则旨在保证某些下降算法的全局收敛性，我们引入归一化的Goldstein型搜索规则并将其与局部极小极大值方法在数值实现和理论分析上都适合于发现半线性椭圆型偏微分方程的多个不稳定解。与最初在[Y. Li和J. Zhou，SIAM J. Sci。计算 ，24（3）：865–885，2002]，然后在[ZQ谢，袁玉杰和周J.暹罗科学 计算 ，34（1）：A395-A420，2012]，我们的方法可以防止步长自动过小，然后通过充分利用两个不等式确保迭代取得合理的进展。NG-LMM的可行性得到了严格的验证。此外，NG-LMM的全局收敛性是在假设峰选择仅是连续的弱假设下得到严格证明的。最后，它被实现为解决多个不稳定解在正方形或哑铃形域上的几个典型的半线性椭圆形边值问题，数值结果表明该方法表现良好。