M-duality in fixed-point problems for general normed spaces

Abstract

Solving fixed-point problems by reducing residual bounds is one of the central approaches in fixed-point iteration, and, in particular, the Halpern iteration is known to effectively reduce the residual. Recently, a new type of duality theorem has been discovered in the convex optimization and fixed-point problem literature and establish a certain symmetry in the convergence behavior of algorithms that are dual to each other. However, these duality theorems rely on a Lyapunov-style proof structure and mostly focus on Hilbert spaces. In this work, we first establish an M-duality theorem for Mann iterations with Lipschitz operators in general normed spaces, assuming only mild conditions on the coefficients of the iterations. Moreover, leveraging this M-duality, we derive new algorithms that attain a strictly smaller worst-case residual bound than that of the Halpern iteration, which was previously known as the state-of-the-art method in general normed spaces.