In this paper, by designing a normalized nonmonotone search strategy with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang--Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai--Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai--Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.

翻译：本文通过设计结合Barzilai-Borwein型步长的归一化非单调搜索策略，提出并分析了一种新型局部极小化极大方法（LMM）——即全局收敛的迭代方法——用于寻找Hilbert空间中非凸泛函的多个（不稳定）鞍点。与采用单调搜索策略的传统LMM相比，本方法不要求每次迭代步中目标泛函值严格递减，从而以更少的计算量实现更快的收敛。首先，基于耦合局部峰值选取（将迭代点拉回解子流形）的归一化迭代格式，通过将优化理论中的Zhang-Hager（ZH）搜索策略推广至LMM框架，引入一种归一化ZH型非单调步长搜索策略，进而构建新型非单调LMM。在放宽泛函在迭代序列上单调性的条件下，严格论证了该方法的可行性与全局收敛性。其次，为加速非单调LMM收敛，通过显式构造Barzilai-Borwein型步长作为每次迭代中归一化ZH型非单调步长搜索策略的试验步长，提出全局收敛的Barzilai-Borwein型LMM（GBBLMM）。最后，将GBBLMM算法应用于两类具有变分结构的半线性椭圆边值问题的多不稳定解求解：其一是齐次Dirichlet边界条件的半线性椭圆方程，其二是半线性Neumann边界条件的线性椭圆方程。大量数值结果表明，该方法高效显著，且大幅加速了LMM的收敛。