Nonlinear three-operator splitting algorithms with momentum for monotone inclusions

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.