Nonmonotone Local Minimax Methods for Finding Multiple Saddle Points (Journal of Computational Mathematics, 2024)
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) saddlepoints of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs withmonotone search strategies, this approach, which does not require strict decrease of theobjective functional value at each iterative step, is observed to converge faster with lesscomputations. Firstly, based on a normalized iterative scheme coupled with a local peakselection that pulls the iterative point back onto the solution submanifold, by generalizingthe 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, andthen a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functionalat the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai-Borwein-type LMM (GBBLMM) is presentedby explicitly constructing the Barzilai-Borwein-type step-size as a trial step-size of thenormalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, theGBBLMM algorithm is implemented to find multiple unstable solutions of two classes ofsemilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and anotheris the linear elliptic equations with semilinear Neumann boundary conditions. Extensivenumerical results indicate that our approach is very effective and speeds up the LMMssignificantly.
Keywords: Multiple saddle points; Local minimax method; Barzilai-Borwein gradient method; Normalized nonmonotone search strategy; Global convergence.