An enriched second-order method for nonconvex composite sparse optimization problems

TL;DR

This paper introduces a novel second-order optimization method tailored for nonconvex composite sparse problems, leveraging second-order information and projection techniques to improve solution efficiency across various applications.

Contribution

It extends the OESOM algorithm to a full second-order method for linear composite $ ext{l}_1$ problems, applicable to a broad class of nonconvex sparse optimization tasks.

Findings

Demonstrates efficiency in computational experiments

Applicable to differential graph operator problems

Outperforms existing methods in selected benchmarks

Abstract

In this paper we propose a second--order method for solving \emph{linear composite sparse optimization problems} consisting of minimizing the sum of a differentiable (possibly nonconvex function) and a nondifferentiable convex term. The composite nondifferentiable convex penalizer is given by $\ell_1$--norm of a matrix multiplied with the coefficient vector. The algorithm that we propose for the case of the linear composite $\ell_1$ problem relies on the three main ingredients that power the OESOM algorithm \cite{dlrlm07}: the minimum norm subgradient, a projection step and, in particular, the second--order information associated to the nondifferentiable term. By extending these devices, we obtain a full second--order method for solving composite sparse optimization problems which includes a wide range of applications. For instance, problems involving the minimization of a general class \emph{differential graph operators} can be solved with the proposed algorithm. We present several computational experiments to show the efficiency of our approach for different application examples.