A nonmonotone globalization framework for Anderson acceleration for contractive and nonexpansive fixed point problems

Abstract

Anderson acceleration (AA) is an effective technique for accelerating fixed point iterations, but it generally lacks global convergence guarantees. We propose a nonmonotone globalized Anderson acceleration framework that retains the local acceleration of AA while ensuring global convergence. For contractive mappings, without requiring prior knowledge of the contraction factor, we prove that the proposed method reduces exactly to pure AA after finitely many iterations and enjoys global $r$-linear convergence for memory size $m\geq1$ and global residual $q$-linear convergence for $m=1$, with convergence factors no greater than the contraction factor of the underlying fixed point mapping. For nonexpansive mappings, we prove that the fixed point residuals converge globally to zero. A class-agnostic parameter selection rule is further developed to ensure these convergence properties in both settings. Numerical experiments demonstrate the robustness and efficiency of the proposed method.