Nonconvex flexible sparsity regularization: theory and monotone numerical schemes

TL;DR

This paper introduces a new nonconvex regularization method for sparse solutions in operator equations, providing theoretical convergence analysis and demonstrating advantages of variable penalties through numerical experiments.

Contribution

It develops a general nonconvex regularization framework with convergence guarantees and proposes monotone algorithms, highlighting benefits of variable penalties over fixed ones.

Findings

Convergence of the proposed regularization method is established.

Monotone algorithms effectively solve nonconvex problems.

Variable penalties outperform fixed penalties in numerical tests.

Abstract

Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and numerical perspectives. Namely, we show convergence of the regularization method and establish convergence properties of a couple of majorization approaches for the associated nonconvex problems. We also test a monotone algorithm for an academic example where the operator is an $M$ matrix, and on a time-dependent optimal control problem, pointing out the advantages of employing variable penalties over a fixed penalty.