腾讯会议：667-764-287 密码：202207
邀请人：陈金如, 王锋
摘要： In this talk, combining the normalized nonmonotone search strategy with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is globally convergent, is proposed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs, this approach does not require the strict decrease of the objective functional value at each iterative step. Firstly, by generalizing the Zhang--Hager (ZH) search strategy in the optimization theory, a kind of normalized nonmonotone step-size search strategy is introduced to traditional LMMs, and the nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out. Secondly, in order to speed up the convergence of the nonmonotone LMMs, 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 nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM 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 LMM significantly.