Iterative Proximal-Minimization for Computing Saddle Points with Fixed Index

TL;DR

This work introduces Iterative Proximal Minimization (IPM) to compute saddle points with fixed Morse index by augmenting the Iterative Minimization Formulation with a proximal penalty $\rho d(x,y)$ that grows faster than quadratic. It establishes a differential-game interpretation in which the Nash equilibrium of a three-agent game corresponds to a saddle point of the potential $V$, and shows that a suitably chosen proximal term yields well-posed subproblems and improved robustness without increasing per-iteration cost. The authors prove convergence properties, provide a generalization to index-$k$ saddle points using the lowest $k$ eigenvectors, and demonstrate the method on a 2D ODE example, the Cahn–Hilliard equation, and the Allen–Cahn equation, highlighting regimes where larger $\rho$ expands basins of attraction and increases robustness at the expense of efficiency. The practical impact lies in a more reliable, parameter-tunable framework for locating transition states on complex energy landscapes, with implications for modeling noise-induced transitions in physics and chemistry. Overall, IPM offers a principled, game-theoretic augmentation of IMF that enhances convergence stability and broadens applicability to high-dimensional, non-convex systems.

Abstract

Computing saddle points with a prescribed Morse index on potential energy surfaces is crucial for characterizing transition states for nosie-induced rare transition events in physics and chemistry. Many numerical algorithms for this type of saddle points are based on the eigenvector-following idea and can be cast as an iterative minimization formulation (SINUM. Vol. 53, p.1786, 2015), but they may struggle with convergence issues and require good initial guesses. To address this challenge, we discuss the differential game interpretation of this iterative minimization formulation and investigate the relationship between this game's Nash equilibrium and saddle points on the potential energy surface. Our main contribution is that adding a proximal term, which grows faster than quadratic, to the game's cost function can enhance the stability and robustness. This approach produces a robust Iterative Proximal Minimization (IPM) algorithm for saddle point computing. We show that the IPM algorithm surpasses the preceding methods in robustness without compromising the convergence rate or increasing computational expense. The algorithm's efficacy and robustness are showcased through a two-dimensional test problem, and the Allen-Cahn, Cahn-Hilliard equation, underscoring its numerical robustness.

Figures (5)

• Figure 1: Decay of errors (measured by $\|\nabla V(x^t)\|$) in the iterative proximal minimization scheme, where $x$-axis is the number of iterations $k$. $M$ is the steps of gradient descent in the subproblem for $\widetilde{W}_\rho$ (see Algorithm \ref{['alg:algorithm1']}).
• Figure 2: Comparison of attraction basins towards each of three saddle points when $\rho$ varies. Each color corresponds to the basin of each saddle point. The index-1 region $\Omega_1$ of the function $V$ is shown in the first panel.
• Figure 3: Results by adding the penalty term: $\rho |\phi -\phi^l|^4 (b=4):$ (a) Transition state (solid curves) and the initial state (dashed line); (b) The decay of the error $\| \delta_\phi F(\phi^{l})\|_{L^2}$ with the outer cycle for fixed inner iteration number $M=600$ by using various $\rho = 0.3, 1, 5, 10, 20.$ (c) The decay of the error $\| \delta_\phi F(\phi^{l})\|_{L^2}$ with the outer cycle when $\rho=0.3$(solid lines) and $\rho=0$ (dashed lines), respectively, by using various inner iteration number 100, 200, 400 and 1272. The case for $M > 450$ with $\rho=0$ is divergent; (d) The decay of the error $\| \delta_\phi F(\phi^{l})\|_{L^2}$ with the outer cycle for fixed inner iteration number $M=10^4$ by using various $\rho = 1, 5, 10, 20.$ (e) The decay of the error $\| \delta_\phi F(\phi^{l})\|_{L^2}$ with the outer cycle when $\rho=10$(solid lines) and $\rho=0$ (dashed lines), respectively, by using various inner iteration number $0.5\times 10^4, 1.0\times 10^4$ and $1.5\times 10^4$. The case for $M > 450$ with $\rho=0$ is divergent.
• Figure 4: Comparison of attraction basins towards each of three saddle points when $\rho$ varies, for a cubic penalty function $d(x,y) = \sum_{i=1}^d\|x_i-y_i\|^3$. With the same value of $\rho$, we have a larger attraction basin with the cubic penalty function, compared with quartic penalty function (Fig. \ref{['fig:attraction_basin']}).
• Figure 5: (a) The decay of the error $\| \delta_\phi F(\phi^{l})\|_{L^2}$ with the outer cycle for fixed inner iteration number $M=10^4$ and with the penalty $\rho |\phi-\phi^l|^3$ (i.e., b=3), by using various $\rho = 0.4, 1, 5, 10, 20.$ (b) The comparison of the error decay for $b=3$ (solid lines) and $b=4$ (dashed lines) when using various $\rho=0.4, 1, 5$. Here, the inner iteration number is fixed at $M=10^4$. The case for $b=4$ when $\rho<0.66$ is divergent. (c) The comparison of the error decay for $b=3$ (solid lines) and $b=4$ (dashed lines) when using various inner iteration number $M=10^3$ and $2 \times 10^4$. Here $\rho=0.4$ is fixed and the case for $b=4$ with $M > 10^3$ is divergent.

Theorems & Definitions (19)

• Definition 1: Pure Nash equilibrium
• Remark 1
• Definition 2
• Theorem 1
• Lemma 2
• Proof 1
• Theorem 3
• Proof 2
• Remark 2
• Proof 3: Proof of Theorem \ref{['lem:fs']}