DC algorithms for a class of sparse group $\ell_0$ regularized optimization problems

TL;DR

This paper introduces a continuous relaxation model for sparse group  regularized optimization, proves its equivalence to the original problem, and develops DC algorithms with convergence guarantees and practical efficiency.

Contribution

It establishes a novel relaxation model for sparse  regularized problems and designs DC algorithms with proven convergence and finite support recovery.

Findings

Algorithms converge to stationary points with shared support.

Finite iteration support recovery is achieved.

Numerical experiments demonstrate efficiency.

Abstract

In this paper, we consider a class of sparse group $\ell_0$ regularized optimization problems. Firstly, we give a continuous relaxation model of the considered problem and establish the equivalence of these two problems in the sense of global minimizers. Then, we define a class of stationary points of the relaxation problem, and prove that any defined stationary point is a local minimizer of the considered sparse group $\ell_0$ regularized problem and satisfies a desirable property of its global minimizers. Further, based on the difference-of-convex (DC) structure of the relaxation problem, we design two DC algorithms to solve the relaxation problem. We prove that any accumulation point of the iterates generated by them is a stationary point of the relaxation problem. In particular, all accumulation points have a common support set and a unified lower bound for the nonzero entries, and their zero entries can be attained within finite iterations. Moreover, we prove the convergence of the entire iterates generated by the proposed algorithms. Finally, we give some numerical experiments to show the efficiency of the proposed algorithms.