Nonlinear three-operator splitting algorithms with momentum for monotone inclusions

TL;DR

This paper presents three innovative nonlinear splitting algorithms with momentum for solving structured monotone inclusion problems, demonstrating their convergence and efficiency through theoretical analysis and numerical experiments.

Contribution

The paper introduces three novel momentum-augmented splitting algorithms for monotone inclusions, extending classical schemes with convergence guarantees.

Findings

Algorithms converge weakly under step-size conditions.

Strong monotonicity yields R-linear convergence.

Numerical tests show superior performance on portfolio optimization problems.

Abstract

In this paper, we introduce three novel splitting algorithms for solving structured monotone inclusion problems involving the sum of a maximally monotone operator, a monotone and Lipschitz continuous operator and a cocoercive operator. Each proposed method extends one of the classical schemes: the semi-forward-reflected-backward splitting algorithm, the semi-reflected-forward-backward splitting algorithm, and the outer reflected forward-backward splitting algorithm by incorporating a nonlinear momentum term. Under appropriate step-size conditions, we establish the weak convergence of all three algorithms, and further prove their $R$-linear convergence rates under strong monotonicity assumptions. Preliminary numerical experiments on both synthetic datasets and real-world quadratic programming problems in portfolio optimization demonstrate the effectiveness and superiority of the proposed algorithms.