Inexact Warped Resolvent iterations

Abstract

In this paper we aim to solve structured monotone inclusions using inexact warped resolvents evaluated under a relative-error criterion. The resulting algorithms admit a geometric interpretation as relaxed projection methods onto dynamically generated cuts, extending classical projection--proximal and hybrid extragradient proximal frameworks to nonlinear warped resolvent. Under mild assumptions, we establish weak convergence of the iterates. We further derive strong convergence results by incorporating projection steps onto intersections of halfspaces via Haugazeau-type scheme, as well as linear convergence under a metric subregularity assumption. The proposed algorithm provides a unified framework for incorporating inexact resolvent computations into several classical schemes arising in monotone operator theory and primal-dual optimization, such as Tseng's forward-backward-forward splitting, forward-backward-half-forward, Chambolle--Pock, and Condat--Vũ. Finally, we present applications in saddle-point and structured convex minimization problems. Numerical experiments on synthetic saddle-point instances and computed tomography reconstruction demonstrate the computational advantages of the proposed methods.