Nonlinear three-operator splitting algorithms with momentum for monotone inclusions
Liqian Qin, Aviv Gibali, Cuijie Zhang, Yuchao Tang
TL;DR
This work addresses monotone inclusion problems of the form $0 \in Ax+Bx+Cx$ by introducing three nonlinear momentum-augmented splitting schemes that leverage warped resolvents to exploit problem structure. The methods—nonlinear SFRBS, SRFBS, and ORFBS—are shown to converge weakly under standard assumptions, with $R$-linear convergence when $A$ is strongly monotone. The paper provides rigorous convergence analyses and linear rates, plus extensive numerical experiments on synthetic data and portfolio-optimization quadratic programs confirming faster convergence and robustness compared with existing schemes. These results broaden the toolbox for multi-operator splitting, enabling more efficient solutions for structured monotone inclusions and suggesting avenues for four-operator extensions.
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.