Chunxiang Zong, Xiaolong Qin, A fast three-operator splitting method with $o(1/k)$ last-iterate convergence rates for monotone inclusions

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DOI: 10.23952/jano.8.2026.1.03
Volume 8, Issue 1, 1 April 2026, Pages 19-54

Abstract. In this paper, we introduce a fast three-operator splitting algorithm to solve a monotone inclusion problem with the sum of a maximally monotone operator, a monotone Lipschitz-continuous operator, and a cocoercive operator in a real Hilbert space. Under mild assumptions, we establish the weak convergence of the generated iterative sequence and the convergence rate of $o(1/k)$ for both the discrete velocity and the tangent residual at the last iterate. To apply the proposed fast algorithm to minimize the sum of three functions with a linear operator, we develop a fully splitting primal-dual algorithm that provably converges to a solution while attaining an $o(1/k)$ last-iterate convergence rate for the discrete velocity, the tangent residual and primal-dual gap. Finally, we present numerical experiments to demonstrate the effectiveness and competitive performance of our proposed algorithm compared to existing methods in the literature.

How to Cite this Article:
C. Zong, X. Qin, A fast three-operator splitting method with $o(1/k)$ last-iterate convergence rates for monotone inclusions, J. Appl. Numer. Optim. 8 (2026), 19-54.