Generalized Forward-Backward Splitting with Penalization for Monotone Inclusion Problems

TL;DR

This paper presents a new generalized forward-backward splitting method with penalization for solving complex monotone inclusion problems, demonstrating convergence properties and practical effectiveness through numerical experiments.

Contribution

It introduces a novel splitting algorithm with penalty terms for monotone inclusions, establishing convergence results and applying it to large-scale hierarchical convex optimization problems.

Findings

Weak ergodic convergence to solutions

Strong convergence under strong monotonicity

Effective in large-scale convex minimization tasks

Abstract

We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximal monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided the condition corresponded to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a large-scale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.