NONMONOTONE LOCAL MINIMAX METHODS FOR FINDING MULTIPLE SADDLE POINTS  

作　　者：Wei Liu Ziqing Xie Wenfan Yi 

机构地区：[1]South China Research Center for Applied Mathematics and Interdisciplinary Studies,South China Normal University,Guangzhou 510631,China [2]Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education),Hunan Normal University,Changsha,Hunan 410081,China [3]Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics,School of Mathematics,Hunan University,Changsha,Hunan 410082,China

出　　处：《Journal of Computational Mathematics》2024年第3期851-884,共34页计算数学（英文）

基　　金：supported by the NSFC(Grant Nos.12171148,11771138);the NSFC(Grant Nos.12101252,11971007);the NSFC(Grant No.11901185);the National Key R&D Program of China(Grant No.2021YFA1001300);by the Fundamental Research Funds for the Central Universities(Grant No.531118010207).

摘　　要：In this paper,by designing a normalized nonmonotone search strategy with the BarzilaiBorwein-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.

关 键 词：Multiple saddle points Local minimax method Barzilai-Borwein gradient method Normalized nonmonotone search strategy Global convergence 

分 类 号：O24[理学—计算数学]