An accelerated rearrangement method for two-phase composite optimization

TL;DR

This paper introduces an Accelerated Rearrangement Method (ARM) for nonconvex two-phase composite optimization, improving convergence rates over classical methods through momentum-like acceleration and restart strategies, with demonstrated efficiency in numerical experiments.

Contribution

The paper develops a novel accelerated rearrangement algorithm with proven convergence and improved rates, extending classical methods for nonconvex composite optimization problems.

Findings

ARM converges faster than classical rearrangement methods.

Numerical experiments confirm practical efficiency in 2D and 3D.

Asymptotic convergence rates are improved in one dimension.

Abstract

We propose and analyze an Accelerated Rearrangement Method (ARM) for solving a class of nonconvex optimization problems involving two-phase composites. These problems include maximizing the (work) energy of a membrane governed by the Poisson equation and minimizing the principal eigenvalue of a weighted Dirichlet-Laplacian, both subject to material distribution constraints. Building on the classical rearrangement method, we introduce momentum-like acceleration by extrapolating the Fr\'echet derivative, leading to a provably convergent algorithm. We also introduce a restarted variant that guarantees monotonic improvement of the objective. In one dimension, we derive asymptotic convergence rates for ARM and prove that they improve upon the classical rearrangement method. Numerical experiments in both two and three dimensions confirm the accelerated convergence and demonstrate practical efficiency.