  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 [Y. Li and J. Zhou, SIAM J. Sci. Comput.,
24(3), 865--885 (2002)] and play 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 framework named as the normalized Wolfe-Powell-type LMM
(NWP-LMM) is 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 NWP-LMM algorithm. 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 than its known LMM companions.
