Nonlinear Spectral Duality

TL;DR

This paper explores nonlinear spectral duality for convex functions, showing how to transform nonlinear eigenvalue problems into dual formulations that preserve key spectral properties, aiding in solving complex optimization problems.

Contribution

It introduces a novel framework for nonlinear spectral duality that maintains spectral properties across primal and dual formulations, applicable to various optimization problems.

Findings

Spectral duality preserves spectrum and multiplicities.

Dual formulations can simplify complex nonlinear eigenvalue problems.

Applications demonstrated in graph optimization and convex geometry.

Abstract

Nonlinear eigenvalue problems for pairs of homogeneous convex functions are particular nonlinear constrained optimization problems that arise in a variety of settings, including graph mining, machine learning, and network science. By considering different notions of duality transforms from both classical and recent convex geometry theory, in this work we show that one can move from the primal to the dual nonlinear eigenvalue formulation maintaining the spectrum, the variational spectrum as well as the corresponding multiplicities unchanged. These nonlinear spectral duality properties can be used to transform the original optimization problem into various alternative and possibly more treatable dual problems. We illustrate the use of nonlinear spectral duality in a variety of example settings involving optimization problems on graphs, nonlinear Laplacians, and distances between convex bodies.