Duality Perspective on Nonlinear Eigenproblems

TL;DR

The paper addresses nonlinear eigenproblems for subdifferentials of convex functionals in reflexive Banach spaces by formulating a Fenchel-duality-based framework. It establishes a rigorous primal–dual equivalence between $p$-eigenvectors of $J$ and $q$-eigenvectors of $J^*$, introduces the dual Rayleigh quotient and a duality gap, and interprets the dual problem as the eigenproblem of the inverse operator. The authors develop an inverse-power method (IPM) as a dual power method and prove convergence to nonlinear eigenfunctions, complemented by a proximal-power method interpretation. Numerical experiments on the $p$-Laplacian validate the theory, show monotone improvement of the dual RQ, and demonstrate effective computation of ground-state and higher-order eigenfunctions, with two approaches for higher-order modes: mean-value discretization-based IPM and a geometry-driven cosine-similarity optimization. The work unifies theory and computation for nonlinear spectral problems in Banach spaces and provides practical tools for applications in imaging, clustering, and PDE-based modeling, while highlighting directions for robust higher-order eigenfunction computation.

Abstract

We investigate nonlinear eigenproblems for a broad class of proper, closed, convex functionals in reflexive Banach spaces. We develop a dual formulation of the nonlinear eigenproblem using the Fenchel conjugate and establish an equivalence to the primal problem. Further, we introduce a duality gap and a geometric characterization of eigenvectors that apply in general Banach spaces. We interpret the dual problem as the eigenproblem for the inverse operator of the primal problem. Concerning numerical methods for solving nonlinear eigenproblems, we analyze the inverse power method, framed as a dual power method, showing strong convergence in the case of absolutely p-homogeneous functionals. Our theoretical results are validated by extensive numerical experiments for the p-Laplacian. We further connect the flow-based proximal power method from the literature to the inverse power method and discuss two numerical approaches to approximate higher-order nonlinear eigenfunctions.

Figures (10)

• Figure 1: Sketch of the geometric characterization of nonlinear eigenvectors. The gray ellipses depict the level sets of $J$ and the orange circles the level sets of $H$, which are balls around the origin. There are two vectors $u,v$ with the gradients of $H$ (orange) and $J$ (black) drawn at these points. The gradients are perpendicular to the level sets. The vector $v$ cannot be a an eigenvector since $\nabla H(v)$ and $\nabla J(v)$ do not share the same direction, thus $\nabla J(v)$ is not normal to the balls around the origin. The gradients $\nabla H(u)$ and $\nabla J(u)$ at the point $u$ share the same direction. Indeed, the vector $u$ is an eigenvector if its norm fulfills the condition $|u|_H = \frac{pJ(u)}{|\zeta|_{H^*}}$.
• Figure 2: Convergence plots of the metrics of the inverse power method for different values of $p$. The $x$-axis shows the number of iterations and the $y$-axis depicts the respective value of the metric. We remark that the lower three plots use a logarithmic $y$-axis. From top to down: The dual Rayleigh quotient is shown to be monotonously increasing with respect to the iterates of the inverse power method. $\operatorname{Cosim}$ is the cosine similarity, which becomes $1$ iff the iterate is a $p$-Laplace eigenvector, else it is less than one. Next, we illustrate the duality gap, which becomes zero if the iterate is an eigenvector and otherwise positive. Finally, we plot the $l^2$-error of the iterates fulfilling the eigenvector condition, which is zero iff the iterate is an eigenvector.
• Figure 3: Initial guess (top left) and computed eigenvector for different values of $p$, which is the ground state of the $p$-Laplacian on an L-shaped domain.
• Figure 4: Evaluation of the power method for higher-order eigenfunctions. We evaluate the scheme with the same three metrics as in \ref{['subsect:num_val_ipm']}. The four plots depict the Rayleigh quotient value (top), the cosine similarity (upper center), duality gap (lower center), and $l^2$-error (bottom) of the iterates at every iteration. In the top plot, we can see that our proposed mean value approximation approach converges faster and more accurate than the finite element approaches bobkov2025inverseiterationmethodhigher to the second smallest eigenfunction without plateauing. Furthermore, the 'crossed' mesh, which yields a better domain tesselation, has one plateau less than the 'right' and 'left' mesh. The values marked with '$x$' are the iterates which are local maxima of the cosine similarity and local minima of the duality gap at the same time. These iterates are plotted in \ref{['fig:local_minima_fl']}, \ref{['fig:local_minima_fr']}, and \ref{['fig:local_minima_fc']}.
• Figure 5: Top left: Mesh discretization of the finite element approximation. The remaining plots show the iterates marked in \ref{['fig:higher_order_conv']}, which are the local extrema of the duality gap and cosine similarity of the iteration scheme. The first extremum coincides with the extremum of the mean value approximation. The second extremum only occurs in the 'left' and 'right' discretization. Also the first plateau appears only for the 'left' and 'right' discretization. The third extremum is the final computed eigenfunction, which gets rotated by the algorithm due to the geometry of the domain and the finite elements by 90 degrees.

Theorems & Definitions (41)

• Definition 1: $p$-eigenvector
• Example 1
• Example 2
• Definition 2: Rayleigh quotient
• Definition 3
• Definition 4
• Lemma 1
• proof
• Remark 1
• Definition 5: Dual Rayleigh quotient