Nonmonotone local minimax methods for finding multiple saddle points

TL;DR

This work develops nonmonotone local minimax methods for finding multiple unstable saddle points of nonconvex functionals in Hilbert spaces. It introduces a normalized Zhang--Hager-type nonmonotone step-size within the local minimax framework and augments it with a globally convergent Barzilai--Borwein (BB) strategy (GBBLMM) to accelerate convergence. Theoretical results establish feasibility and global convergence under relaxed monotonicity, and extensive PDE experiments (Dirichlet and nonlinear Neumann problems) demonstrate significant speedups and the ability to locate multiple saddle points with prescribed Morse indices. The approach provides efficient, scalable tools for variational problems in nonlinear PDEs and related applications.

Abstract

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.

Figures (7)

• Figure 5.1: Profiles of ten solutions of the Lane-Emden equation on $\Omega=(-1,1)^2$.
• Figure 5.2: Numerical comparison of the GBBLMM with traditional LMMs in terms of the convergence rate for computing solutions in Fig. \ref{['fig:LE10sols']}: (a) $\sim$ (j) for $u_1\sim u_{10}$, respectively. The horizontal and vertical coordinates represent the number of iterations and the norm of the gradient, respectively.
• Figure 5.3: The changes of the relative energy functional values $E(w_k)-E(u)$ (in logarithmic scale) with respect to the number of iterations $k$ for the GBBLMM and traditional LMMs for computing the solutions $u=u_7$ (left) and $u=u_{10}$ (right) in Fig. \ref{['fig:LE10sols']}.
• Figure 5.4: Profiles of ground state solutions of the Hénon equation on $\Omega=(-1,1)^2$ with different $\ell s$.
• Figure 5.5: Profiles of twelve solutions of the Hénon equation with $\ell=6$ on $\Omega=(-1,1)^2$.

Theorems & Definitions (23)

• Definition 2.1: XYZ2012SISC
• Lemma 2.1: XYZ2012SISC
• Lemma 2.2: LXY2021CMS
• Lemma 2.3: LXY2021CMS
• Lemma 2.4: LXY2021CMS
• Theorem 2.1
• Definition 2.2: Rabinowitz1986
• Theorem 2.2
• Lemma 2.5: XYZ2012SISC
• Lemma 2.6