On a monotone scheme for nonconvex nonsmooth optimization with applications to fracture mechanics

TL;DR

This paper introduces a monotone scheme for solving a broad class of nonconvex, nonsmooth optimization problems with applications in fracture mechanics, demonstrating its effectiveness and comparing it with existing algorithms.

Contribution

It proposes a novel monotonically convergent algorithm for nonconvex nonsmooth optimization problems involving concave compositions, with applications to fracture mechanics.

Findings

Successfully applied to fracture mechanics problems

Outperforms two alternative algorithms in tests

Effective in modeling nonsmooth nonconvex energies

Abstract

A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.