On the monotone and primal-dual active set schemes for $\ell^p$-type problems, $p \in (0,1]$

TL;DR

This paper introduces monotone and primal-dual active set algorithms for solving nonsmooth, nonconvex p-norm optimization problems, providing convergence analysis, optimality conditions, and demonstrating effectiveness through diverse numerical applications.

Contribution

It develops a new combined monotone and primal-dual active set algorithm for p-norm problems, including convergence analysis and practical implementation details.

Findings

Algorithms effectively solve p-norm problems in various applications

Numerical tests show good convergence and solution quality

Applicable to optimal control, fracture mechanics, and image reconstruction

Abstract

Nonsmooth nonconvex optimization problems involving the $\ell^p$ quasi-norm, $p \in (0, 1]$, of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction.