Nonlinear forward-backward-half forward splitting with momentum for monotone inclusions

TL;DR

This paper introduces a novel nonlinear momentum-augmented splitting algorithm for structured monotone inclusion problems, achieving convergence guarantees and demonstrating superior performance in portfolio optimization tasks.

Contribution

It proposes a new nonlinear momentum-based splitting algorithm with convergence analysis and extends it to stochastic variance-reduced versions for finite-sum problems.

Findings

Proposed algorithm converges weakly under appropriate conditions.

Achieves linear convergence under strong monotonicity.

Numerical experiments show improved performance over existing methods.

Abstract

In this work, we propose a new splitting algorithm for solving structured monotone inclusion problems composed of a maximally monotone operator, a maximally monotone and Lipschitz continuous operator and a cocoercive operator. Our method augments the forward-backward-half forward splitting algorithm with a nonlinear momentum term. Under appropriate conditions on the step-size, we prove the weak convergence of the proposed algorithm. A linear convergence rate is also obtained under the strong monotonicity assumption. Furthermore, we investigate a stochastic variance-reduced forward-backward-half forward splitting algorithm with momentum for solving finite-sum monotone inclusion problems. Weak almost sure convergence and linear convergence are also established under standard condition. Preliminary numerical experiments on synthetic datasets and real-world quadratic programming problems in portfolio optimization demonstrate the effectiveness and superiority of the proposed algorithm.