Analysis of the Three-operator Davis-Yin Splitting in the Inconsistent Case

Abstract

This thesis analyzes the Davis–Yin three-operator splitting method in the inconsistent case, where the underlying monotone inclusion problem may fail to have a solution. The Davis–Yin algorithm extends the Douglas–Rachford and forward–backward splitting methods and is effective in reformulating optimization and inclusion problems as fixed-point iterations. Our study investigates its behavior when no fixed point exists. We prove, under mild assumptions, that the Davis–Yin shadow sequence converges to a solution of the normal problem, which represents a minimal perturbation of the original formulation.