Nonlinear forward-backward-half forward splitting with momentum for monotone inclusions
Liqian Qin, Yuchao Tang, Jigen Peng
TL;DR
The paper addresses solving structured monotone inclusions of the form $0\in Ax+Bx+Cx$ by proposing a nonlinear forward-backward-half forward splitting algorithm with momentum, plus a stochastic variance-reduced variant for finite-sum problems. It establishes weak convergence and linear convergence under strong monotonicity, and proves weak almost sure convergence with linear rates for the VR variant. The authors provide a comprehensive convergence analysis using Lyapunov-type functions and operator-theoretic assumptions, and validate the methods with numerical experiments on nonlinear constrained optimization and portfolio optimization problems. The work yields a scalable, fully splitting framework that leverages the distinct properties of each operator, with practical impact for large-scale monotone inclusions in signal processing, imaging, and optimization contexts.
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.