On proximal point method with degenerate preconditioner: well-definedness and new convergence analysis

Abstract: We study the basic properties of degenerate preconditioned resolvent based on restricted maximal monotonicity, and extend the non-expansiveness, demiclosedness and Moreau's decomposition identity to degenerate setting. Several conditions are further proposed for the well-definedness of the degenerate resolvent and weak convergence of its associated fixed point iterations within either range space or whole space. The results help to understand the behaviours of many operator splitting algorithms, especially in the kernel space of degenerate preconditioner.