The local minimax method (LMM) proposed in [Y. Li and J. Zhou, SIAM J. Sci. Comput. 23(3), 840--865 (2001)] and [Y. Li and J. Zhou, SIAM J. Sci. Comput. 24(3), 865--885 (2002)] is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in the above work and playing a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm named as the normalized Wolfe-Powell-type LMM (NWP-LMM) are introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the LMM algorithms. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM's family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better among its LMM companions.
翻译：在[Y. Li和J. Zhou、SIAM J. Sci. Comput. 23(3)、840--865 (2001)和[Y. Li和J. Zhou、SIAM J. Sci. Comput.24(3)、865-885 (2002)]中提议的当地迷你法是解决非线性椭圆部分方程(PDEs)和多种解决方案的某些变异结构的有效方法。上述工作中采用了最陡峭的下降方向和Armijo型逐步级数级搜索规则,在传统LMMM的性能和趋同性分析中,LMMMM(NWP-LMM)的新算法框架基于一般的下降方向和趋同性能(NWP-LMM)的全球性趋近和趋同,LMMML(ML)类的底值、LMML(ML)级数级程和LG级数级数级数级程(LMML)的基底图和G级底基底图(LMMLF)中,LMMLMLML(ML-L-L-L-L-L-L-L-l-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-l-L-L-L-L-L-l-l-l-l-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-L-
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